Topology is a branch of mathematics that deals with the properties of space that are preserved under continuous transformations, such as stretching, twisting, crumpling, and bending, but not tearing or gluing. It focuses on the concepts of structure, continuity, and convergence, and is often described as "rubber-sheet geometry" because of its emphasis on the flexible and qualitative aspects of geometric forms.
In mathematics, compactification is a technique used to extend a space such that it becomes compact. Compactness is a topological property that has important implications in various areas of mathematics, particularly in analysis and topology. ### General Idea The process of compactification typically involves adding "points at infinity" or otherwise altering the topology of a space to ensure that every open cover of the space has a finite subcover.
The Alexandroff extension is a concept in topology, specifically in the study of topological spaces. It can be seen as a method to extend a given topological space by adding a "point at infinity," thereby creating a new space that retains certain properties of the original.
In topology, the concept of an "end" provides a way to classify the asymptotic behavior of a space at infinity. More formally, an end of a topological space can be understood as a way to describe how the space can be "accessed" from large distances.
In mathematics, particularly in the field of complex analysis, the term "prime end" refers to a concept used in the study of conformal mappings and, more generally, in the theory of Riemann surfaces and potential theory. The notion was introduced by the mathematician Henri Poincaré. **Prime Ends**: Prime ends can be thought of as a way to extend the notion of boundary points in a domain in the complex plane.
The Stone–Čech compactification is a mathematical concept in topology that extends a topological space to a compact space in a way that retains certain properties of the original space. It is named after mathematicians Marshall Stone and Eduard Čech. ### Definition Let \( X \) be a completely regular topological space.
General topology, also known simply as topology, is a branch of mathematics that deals with the properties of space that are preserved under continuous transformations. It provides the foundational language and concepts for many areas of mathematics by introducing notions such as continuity, compactness, connectedness, and convergence without relying on the traditional metrics of distance from geometry. Key concepts in general topology include: 1. **Topological Spaces**: A set equipped with a collection of open subsets that satisfy certain axioms.
Continuum theory is a branch of mathematics that deals with the properties and structures of continua, which can be understood as "continuous" sets. The most common context for discussing continuum theory is in topology, where it often focuses on the study of spaces that are connected and compact, such as the real number line or various types of geometrical shapes.
The term "composant" is French for "component." In various contexts, it refers to a part or element that can be combined with others to form a larger system or structure. Here are some contexts where "composant" might be relevant: 1. **Software Development**: In programming, a "composant" can refer to a reusable software component, such as a module or library that encapsulates functionality.
In topology, a **continuum** refers to a specific type of topological space that is compact, connected, and locally connected. More formally, a continuum is a non-empty, compact, connected space in which every point is part of a connected subset. Here are key properties of a continuum: 1. **Compactness**: This means that every open cover of the space has a finite subcover.
In mathematics, particularly in topology, a **dendrite** is a specific type of topological space that is characterized by a number of distinct features. Here are the key properties and definitions associated with dendrites: 1. **Tree-like Structure**: A dendrite can be thought of as a continuum (a compact, connected metric space) that resembles a tree. It is typically connected and does not contain any loops, which means it is locally tree-like.
Dendroid, in the context of topology, refers to a specific type of topological space that is similar to the structure of a tree but can be generalized in various ways. Generally, a dendroid is a locally connected, compact, non-empty, continuum that is also a dendritic (tree-like) structure. Key characteristics of dendroids include: 1. **Locally Connected**: Every point within a dendroid has a neighborhood base consisting of connected sets.
An indecomposable continuum is a concept from topology, specifically in the study of continua (which are compact, connected metric spaces). A continuum \( X \) is said to be indecomposable if it cannot be represented as the union of two proper, non-empty, closed subsets.
In the context of topology, a **pseudo-arc** is a specific type of continuum. It can be defined as a locally connected, continuum that is irreducible (meaning it cannot be represented as the union of two proper subcontinua) and has the property that any two points in the continuum can be connected by a unique arc.
Separation axioms are a set of conditions in topology that describe how distinct points and sets can be "separated" from each other using open sets. These axioms help to classify topological spaces based on their separation properties. The different separation axioms build upon each other, and they include: 1. **T0 (Kolmogorov)**: A space is T0 if for any two distinct points, there exists an open set containing one of the points but not the other.
In topology, a **Dowker space** is a specific kind of topological space that has peculiar properties related to separability. A space \(X\) is called a Dowker space if it is a normal space (which means that any two disjoint closed sets can be separated by neighborhoods) but not every countable closed set in \(X\) can be separated from a point not in the closed set by disjoint neighborhoods.
A Hausdorff space, also known as a \(T_2\) space, is a type of topological space that satisfies a particular separation property.
The separation axioms are a series of concepts in topology that delineate how distinct points and sets can be separated by open sets. They are integral to the development of topology as a field and have evolved through the contributions of various mathematicians over time.
A Kolmogorov space, also known as a \( T_0 \) space, is a type of topological space that satisfies a specific separation axiom. In a Kolmogorov space, for any two distinct points \( x \) and \( y \), there exists an open set containing one of the points but not the other. This means that for any two points in the space, it is possible to find an open set that "separates" them.
A **locally Hausdorff space** is a topological space in which every point has a neighborhood that is Hausdorff.
In topology, a normal space is a specific type of topological space that satisfies certain separation properties. A topological space \( X \) is called **normal** if it meets the following criteria: 1. **It is a T1 space**: This means that for any two distinct points in the space, there exist open sets that contain one point but not the other. In other words, points can be separated by neighborhoods.
In topology, a **paracompact space** is a topological space with a specific property regarding open covers. A topological space \( X \) is said to be paracompact if every open cover of \( X \) has an open locally finite refinement.
In topology, a **semiregular space** is a type of topological space with specific properties regarding the relationships between open sets and points.
In topology, a **T1 space** (also known as a **Fréchet space**) is a type of topological space that satisfies a particular separation axiom. Specifically, a topological space \( X \) is considered T1 if, for any two distinct points \( x \) and \( y \) in \( X \), there are open sets that separate these points.
Urysohn's lemma is a fundamental result in topology, particularly in the area of general topology dealing with normal spaces.
In topology, the concepts of Urysohn spaces and completely Hausdorff spaces refer to certain separation axioms that describe the ability to distinguish between points and sets within a topological space.
A **Weak Hausdorff space** is a specific type of topological space that extends the usual concept of Hausdorff spaces. In a common Hausdorff space, for any two distinct points, there exist disjoint open sets containing each point. Weak Hausdorff spaces relax this condition, allowing for a certain "closeness" between points.
An **adherent point** is a concept in topology, a branch of mathematics. In the context of a topological space, an adherent point (or limit point) of a set refers to a point that is either in the set itself or is a limit point of that set.
Appert topology is a concept in the field of topology, specifically a type of topology on a set that is defined via a particular collection of open sets. The Appert topology is based on the idea of "approximating" the standard topology of a topological space through certain properties.
The Axiom of Countability is a principle in set theory that deals with the properties of countable sets. In the context of set theory, a set is considered countable if it can be put into a one-to-one correspondence with the set of natural numbers (i.e., it can be enumerated). Specifically, the Axiom of Countability generally refers to the notion that certain mathematical structures possess countable bases or countable properties.
In topology, a **base** for a topological space is a collection of open sets that can be used to generate the topology on that space.
In topology, the concept of a boundary is associated with the idea of the closure and interior of a set in a topological space.
A **Cauchy space** is a concept from the field of topology and analysis, named after the mathematician Augustin-Louis Cauchy. It generalizes certain properties of sequences and convergence in metric spaces, allowing for a more abstract setting in which to study convergence and completeness. In more formal terms, a **Cauchy space** is defined in the following way: 1. **Set and Filter**: Start with a set \( X \).
In topology, a set is called **clopen** if it is both **closed** and **open**. To understand this concept, we need to clarify what it means for a set to be open and closed: 1. A set \( U \) in a topological space is **open** if, for every point \( x \) in \( U \), there exists a neighborhood of \( x \) that is entirely contained within \( U \).
In topology, a **closed set** is a fundamental concept related to the structure of a topological space. A subset \( C \) of a topological space \( X \) is called closed if it contains all its limit points. Here are some important properties and characteristics of closed sets: 1. **Complement**: A set is closed if its complement (with respect to the whole space \( X \)) is open.
In mathematics, "closeness" often refers to a concept related to the distance between points, objects, or values in a particular space. It can be defined in various contexts, such as in metric spaces, topology, and real analysis.
In topology, the **closure** of a set refers to a fundamental concept related to the limit points and the boundary of that set within a given topological space. Specifically, the closure of a set \( A \) in a topological space \( (X, \tau) \) is the smallest closed set that contains \( A \).
Cocountable topology is a specific type of topology defined on a set where a subset is considered open if it is either empty or its complement is a countable set. More formally, let \( X \) be a set. The cocountable topology on \( X \) is defined by specifying that the open sets are of the form \( U \subseteq X \) such that either: 1. \( U = \emptyset \), or 2.
Cofiniteness is a concept often discussed in the context of model theory and formal languages, particularly related to the properties of certain mathematical structures. In general, a property or structure is said to exhibit cofiniteness when the complement set (or the set of elements that do not belong to it) is finite.
Coherent topology is a type of topology that is often used in the context of sheaf theory and algebraic geometry, particularly to study the behavior of sheaves over topological spaces. It is commonly associated with coherent sheaves, which are a particular type of sheaf that can be thought of as a generalization of vector bundles or modules over a ring.
The compact-open topology is a topology defined on the space of continuous functions between topological spaces, particularly when considering the set of continuous functions from one topological space to another. This topology is especially useful in areas like functional analysis and algebraic topology.
The comparison of topologies generally refers to the process of analyzing and contrasting different topological structures on a set. In the context of topology, this involves examining how various topologies can be defined on the same set and how they relate to one another in terms of properties and behavior.
A completely metrizable space is a topological space that can be given a metric (or distance function) such that the topology induced by this metric is the same as the original topology of the space, and furthermore, the metric is complete. To break this down: 1. **Topological Space**: This is a set of points, along with a collection of open sets that satisfy certain axioms (like closure under unions and finite intersections).
A completely uniformizable space is a type of topological space that can be endowed with a uniform structure such that the uniform structure defines a topology that is equivalent to the original topology of the space. In more detail, a uniform space is a set equipped with a filter of entourages that allows us to talk about concepts such as "uniform continuity" and "Cauchy sequences.
In topology, a connected space is a fundamental concept that refers to a topological space that cannot be divided into two disjoint, non-empty open sets. More formally, a topological space \( X \) is called connected if there do not exist two open sets \( U \) and \( V \) such that: 1. \( U \cap V = \emptyset \) 2. \( U \cup V = X \) 3.
A **countably generated space** is a type of topological space that can be described in terms of its open sets. Specifically, a topological space \( X \) is called countably generated if there exists a countable collection of open sets \( \{ U_n \}_{n=1}^\infty \) such that the smallest topology on \( X \) generated by these open sets is the same as the original topology on \( X \).
"Counterexamples in Topology" is a well-known book by Lynn Steen and J. Arthur Seebach Jr. published in 1970. The book is designed as a resource for students and mathematicians to illustrate a wide range of concepts in topology through counterexamples. It motivates the study of topology not only by presenting general theorems and ideas, but also by showing the importance of counterexamples that help clarify the limits of those theorems.
The term "cut point" can refer to different concepts depending on the context, such as mathematics, statistics, and various fields of science and engineering. Here are a few interpretations: 1. **Mathematics/Graph Theory**: In graph theory, a cut point (or articulation point) is a vertex in a graph that, when removed along with its incident edges, increases the number of connected components of the graph.
The Denjoy–Riesz theorem is an important result in real analysis, particularly in the context of functions of a real variable and integration. It deals with the conditions under which a function can be represented as being absolutely continuous and has implications for the behavior of functions that are Lebesgue integrable.
In mathematics, particularly in topology, a set \( A \) is referred to as a **dense set** in a space \( X \) if every point in \( X \) is either an element of \( A \) or is arbitrarily close to a point in \( A \). More formally, a subset \( A \) of a topological space \( X \) is dense in \( X \) if the closure of \( A \) is equal to \( X \).
In mathematics, specifically in the context of topology and set theory, the **derived set** of a given set refers to the set of all limit points (or accumulation points) of that set.
In topology, a branch of mathematics, "development" refers to a concept associated with the way in which a topological space can be represented in terms of more basic or simpler elements. While "development" itself does not have a standard definition in all areas of topology, it is often used in specific contexts dealing with the structure and properties of topological spaces.
In topology, the **disjoint union** (also known as the coproduct in the category of topological spaces) is a way to construct a new topological space from a collection of topological spaces such that the new space captures the "disjointness" of the original spaces.
Double origin topology (also referred to in some contexts as the "double point" space) is a concept in topology that involves a space in which there are two indistinguishable points that serve as 'origins' of the space. This idea can be constructed using set theory and is often used in discussions about defining equivalence classes and understanding the properties of topological spaces, particularly with respect to their connectivity and properties of separation.
The Eberlein compactum is a specific topological space that is an example of a compact space which is not metrizable. It is constructed using the properties of certain compact sets in the space of continuous functions. More formally, an Eberlein compactum can be described as a subspace of the space of all bounded sequences of real numbers, specifically the closed bounded interval [0,1] or some analogous bounded topological space. The compactum is named after the mathematician P.
Either-or topology, also known as the "discrete topology," is a simple kind of topology that can be defined on a set. In this topology, every subset of the set is considered an open set. The discrete topology is characterized by the following properties: 1. **Open Sets**: Every subset of the set is in the topology. This includes the empty set and the entire set itself.
In topology and related fields, an Esakia space is a type of topological space that is associated with the study of certain classes of lattices. Specifically, Esakia spaces arise in the context of modal logic and can be understood in terms of their strong relation to Kripke frames. An Esakia space is characterized by having certain order-theoretic properties that are related to the accessibility relations in modal logic semantics.
Esenin-Volpin's theorem is a result in the field of mathematics, specifically in the area of functional analysis and the theory of distributions. The theorem deals with the relationship between certain types of linear functionals and their representations through measures. The essence of Esenin-Volpin's theorem is that it provides conditions under which a linear functional acting on a space of test functions can be uniquely represented as an integral with respect to a measure.
In topology, a **filter** is a concept used to generalize certain aspects of nets and convergence, particularly in the study of convergence and topological spaces. A filter on a set provides a way to talk about collections of subsets of that set that have certain properties, mainly focusing on "largeness" or "richness" of subsets.
In topology, the concept of a **final topology** (sometimes referred to as the **final topology with respect to a set of maps**) is an important construction that arises particularly in the context of category theory and the study of topological spaces. Intuitively, a final topology is defined in terms of a collection of topological spaces and continuous maps from those spaces to a target space.
Finite topology, often referred to in the context of finite topological spaces, typically involves the study of topological spaces that have a finite number of points. In a finite topological space, the set of points is limited, which leads to simplified structures and properties compared to infinite topological spaces. ### Key Concepts of Finite Topology: 1. **Finite Set**: A finite topological space has a finite number of elements.
In topology, a topological space \( X \) is called *first-countable* if, at every point \( x \in X \), there exists a countable collection of open sets (called a *countable neighborhood basis*) such that any open neighborhood of \( x \) contains at least one of these open sets.
In topology, a Fréchet–Urysohn space is a type of topological space that has a specific property concerning its convergent sequences. A topological space \( X \) is said to be a Fréchet–Urysohn space if, whenever a subset \( A \subseteq X \) is a limit point of a point \( x \in X \), there exists a sequence of points in \( A \) that converges to \( x \).
In topology, a **generic point** is a concept used to describe a point that represents a subset of a topological space in a broad or "generic" sense. Specifically, a point \( x \) in a topological space \( X \) is called a generic point of a subset \( A \) of \( X \) if every open set containing \( x \) intersects \( A \) in a non-empty set.
In the context of topology, a \( G_\delta \) set (pronounced "G delta set") is a subset of a topological space that can be expressed as a countable intersection of open sets.
Half-disk topology is a type of network topology that is used in certain wireless communication systems. It is characterized by a circular or semi-circular arrangement where devices (nodes) are positioned within a half-disk area, facilitating communication among them. In a half-disk topology, the nodes that are placed within the half-disk can communicate directly with one another if they are within range.
"Hedgehog space" is a term that can refer to a couple of different concepts depending on the context, such as mathematics, gaming, or other fields. However, one of the most common references is in topology, particularly in the study of spaces related to the "hedgehog" model in algebraic topology or differential topology.
The Heine–Borel theorem is a fundamental result in real analysis and topology that characterizes compact subsets of Euclidean space. The theorem states that in \(\mathbb{R}^n\), a subset is compact if and only if it is closed and bounded. To elaborate: 1. **Compact Set**: A set \( K \) is compact if every open cover of \( K \) has a finite subcover.
In topology, the concept of an **initial topology** is a way to construct a topology on a set that reflects the structure imposed by a collection of functions (or maps) from that set to other topological spaces. Specifically, it provides a minimal topology that makes certain maps continuous.
The term "integer broom topology" is not a standard term in mathematics or topology, as of my knowledge cut-off in October 2023. However, the concept of a "broom" in topology typically refers to a certain type of space that is designed to illustrate specific properties of convergence and limits.
In topology, the *interior* of a set refers to the collection of all points within that set which are not on its boundary.
Interlocking interval topology is a concept in the field of topology, specifically dealing with spaces constructed using intervals that have a particular relationship with one another. Here's a basic overview of the concept: ### Definitions: 1. **Intervals:** In a typical setting (especially in \(\mathbb{R}\)), intervals can be open, closed, or half-open.
In algebraic geometry, an **irreducible component** of a topological space, particularly a scheme or algebraic variety, is a maximal irreducible subset of that space. To elaborate: 1. **Irreducibility**: A topological space is considered irreducible if it cannot be expressed as the union of two or more nonempty closed subsets.
In topology and mathematical analysis, an **isolated point** (or isolated point of a set) is a point that is a member of a set but does not have other points of the set arbitrarily close to it.
The Katětov–Tong insertion theorem is a result in the field of topology, particularly in the area of set-theoretic topology. It deals with the properties of certain types of topological spaces, specifically separable metric spaces. The theorem is named after mathematicians František Katětov and David Tong.
The Kuratowski–Ulam theorem is a result in the field of topology, specifically within the area of set-theoretic topology. It was conceived by mathematicians Kazimierz Kuratowski and Stanislaw Ulam.
The lexicographic order topology on the unit square, which we denote as \( [0, 1] \times [0, 1] \), is based on an ordering of the points in the unit square. In this topology, we define a way to compare points \((x_1, y_1)\) and \((x_2, y_2)\) in the square using the lexicographic order, similar to how words are ordered in a dictionary.
A **Lindelöf space** is a concept from topology, a branch of mathematics that deals with the properties of space that are preserved under continuous transformations. Specifically, a topological space is termed **Lindelöf** if every open cover of the space has a countable subcover.
In general topology, various examples illustrate different concepts and properties. Here is a list of significant examples that are commonly discussed: 1. **Discrete Topology**: In this topology, every subset is open. For any set \(X\), the discrete topology on \(X\) consists of all possible subsets of \(X\).
General topology, also known as point-set topology, is a branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. Here’s a list of key topics typically covered in a general topology course: 1. **Topological Spaces** - Definition of topological spaces - Basis for a topology - Subspace topology - Product topology - Quotient topology 2.
The term "local property" can refer to different concepts depending on the context in which it is used. Here are a few interpretations of "local property": 1. **Real Estate Context**: In real estate, local property may refer to real estate assets that are situated in a specific geographic area. This can involve considerations like property value, market trends, zoning laws, and community characteristics that pertain to that specific locality.
In topology, a subset \( A \) of a topological space \( X \) is called **locally closed** if it can be expressed as the intersection of an open set and a closed set in \( X \). More formally, a subset \( A \subseteq X \) is locally closed if there exists an open set \( U \subseteq X \) and a closed set \( C \subseteq X \) such that: \[ A = U \cap C.
In topology, a space is said to be **locally connected** at a point if every neighborhood of that point contains a connected neighborhood of that point. More formally, a topological space \(X\) is said to be **locally connected** if for every point \(x \in X\) and every neighborhood \(U\) of \(x\), there exists a connected neighborhood \(V\) of \(x\) such that \(V \subseteq U\).
In topology, a **locally finite space** is a type of topological space that possesses a specific property related to the concept of local finiteness of open covers.
A **mapping torus** is a concept in topology, specifically in the study of fiber bundles and manifolds. It is a way to construct a new topological space from a given manifold and a continuous function defined on it. To describe a mapping torus formally, consider the following: 1. **Space**: Let \( M \) be a topological space (often a manifold) and let \( f: M \to M \) be a continuous map.
In the context of mathematical analysis and topology, a **meagre set** (also known as a **first category set**) is a set that can be expressed as a countable union of nowhere dense sets. A set \( A \) is said to be nowhere dense in a topological space \( X \) if the interior of its closure is empty; that is, there are no open sets in \( X \) that contain any points of \( A \) in a non-empty way.
A **metrizable space** is a topological space that can be endowed with a metric (or distance function) such that the topology induced by this metric is the same as the original topology of the space.
In topology, a **Moore space** is a particular type of topological space that satisfies certain separation axioms and conditions related to bases for open sets. More specifically, a Moore space is a topological space that is a *second-countable* and *reasonable* space.
The Nagata–Smirnov metrization theorem is a fundamental result in topology that provides conditions under which a topological space can be metrized, meaning that the topology of the space can be derived from a metric. This theorem is particularly relevant for spaces that are compact, Hausdorff, and first-countable.
The term "neighbourhood system" can have different meanings depending on the context. Here are a few interpretations: 1. **Urban Planning and Geography**: In urban planning, a neighbourhood system refers to the arrangement and organization of communities within a larger city or metropolitan area. It encompasses residential areas, commercial zones, parks, and public spaces, and focuses on the interactions and relationships between these components.
Nested interval topology is a specific topology defined on the real numbers \(\mathbb{R}\) based on the concept of nested closed intervals. This topology is generated by a base consisting of the sets that can be formulated using nested sequences of closed intervals.
In mathematics, the term "net" can refer to several different concepts, depending on the context. Here are two of the most common interpretations: 1. **Net in Topology**: In topology, a net is a generalization of a sequence that allows the indexing of elements by a directed set. While a sequence is indexed by the natural numbers, a net can be indexed by any directed set, which gives it more flexibility.
It seems there might be a typographical error in your query. If you meant "Node space," "NOC space," or "Nodec Space" in a specific context (like computer networking, mathematics, or some other field), please clarify. As of my last training data, there isn't a widely recognized concept specifically named "Nodec space.
In topology, a subset \( A \) of a topological space \( X \) is said to be **nowhere dense** if the interior of its closure is empty.
In topology, "open" and "closed" maps are concepts that describe certain properties of functions between topological spaces. Here's a brief explanation of each term: ### Open Maps A function \( f: X \rightarrow Y \) between two topological spaces is called an **open map** if it takes open sets in \( X \) to open sets in \( Y \).
In the context of mathematics, particularly in topology, an **open set** refers to a fundamental concept that helps define various properties of spaces. Here's a more detailed explanation: 1. **Definition**: A set \( U \) in a topological space \( X \) is called an open set if, for every point \( x \) in \( U \), there exists a neighborhood around \( x \) that is entirely contained within \( U \).
P-space, or Polynomial Space, is a complexity class in computational complexity theory. It consists of decision problems that can be solved by a deterministic Turing machine using a polynomial amount of memory (space), regardless of the time it takes to compute the answer. In other words, a language belongs to P-space if there exists an algorithm that can decide whether a string belongs to the language using an amount of memory that can be bounded by a polynomial function of the length of the input string.
Parovićenko space, often denoted as \( P \), is a specific type of topological space that is used in the field of general topology. It is particularly interesting because it serves as an example of certain properties and behaviors in topological spaces. The Parovićenko space can be defined as follows: - It is a continuum, meaning it is compact, connected, and Hausdorff.
"Pointclass" is not a widely recognized term in common usage, and it might refer to different things in various contexts. It could pertain to a specific software tool, framework, or concept within a certain field such as programming, data science, or mathematics. For example, in programming contexts, "Pointclass" might refer to a class in object-oriented programming that represents a point in a Cartesian coordinate system, typically containing properties like x and y coordinates.
A Polish space is a concept from the field of topology and descriptive set theory. Specifically, a Polish space is a topological space that is separable (contains a countable dense subset) and completely metrizable (can be endowed with a metric that induces its topology and is complete, meaning every Cauchy sequence converges within the space).
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Just by havin the notion of neighbourhood, concepts such as limit and continuity can be defined without the need to specify a precise numerical value to the distance between two points with a metric.
As an example. consider the orthogonal group, which is also naturally a topological space. That group does not usually have a notion of distance defined for it by default. However, we can still talk about certain properties of it, e.g. that the orthogonal group is compact, and that the orthogonal group has two connected components.