The "infinite broom" is a concept that originated from a visual trick or optical illusion often paired with the idea of an infinite staircase. It can be humorously interpreted or portrayed in various ways, typically involving a broom that appears to endlessly sweep or never run out of bristle length or cleaning capability. In a more abstract or philosophical sense, it might evoke discussions about infinite processes or the nature of infinity in mathematics or philosophy.

The Knaster–Kuratowski fan is a topological space that provides an example of a compact, connected, non-metrizable space. It is constructed to illustrate specific properties in topology, particularly in the context of compactness, connectedness, and the significance of local properties.

Lower limit topology, also known as the standard topology on the real numbers, is a specific topology defined on the set of real numbers \(\mathbb{R}\). This topology is generated by a basis consisting of all half-open intervals of the form \([a, b)\) where \(a < b\).

The Menger sponge is a well-known example of a fractal and a mathematical object that is constructed through an iterative process. It was introduced by the mathematician Karl Menger in 1926. The Menger sponge is defined in three dimensions and is created starting with a solid cube. Here’s how the construction works: 1. **Start with a Cube:** Begin with a solid cube.

In the context of topology, a **nilpotent space** is often associated with the concept of **nilpotent groups** in algebra, particularly in relation to algebraic topology, where one considers the properties of spaces through their homotopy and homology. A topological space is said to be **nilpotent** if its higher homotopy groups become trivial after some finite stage.

In mathematics, a **partially ordered set** (or **poset**) is a set combined with a binary relation that satisfies three properties: reflexivity, antisymmetry, and transitivity. These properties enable us to compare elements of the set in a way that is not necessarily total, meaning not every pair of elements needs to be comparable. 1. **Reflexivity**: For every element \( a \) in the set, \( a \leq a \).

Partition topology is a concept used in the field of topology, specifically in the study of different ways to define topological structures on a set. It involves creating a topology by considering a partition of a set. ### Definitions: - **Set**: A collection of distinct objects, considered as an object in its own right.

A Prüfer manifold, also known as a Prüfer domain or Prüfer ring, is a specific type of mathematical structure studied in commutative algebra and algebraic geometry. It is named after the mathematician Hans Prüfer. In algebraic terms, a Prüfer manifold is a generalized space in which certain sets of ideals exhibit a property similar to that of a Dedekind domain, but with more flexible conditions.

In the context of mathematics, particularly functional analysis and linear algebra, the term "Ran space" typically refers to the range of a linear operator or a linear transformation. The range (or image) of a linear operator \( T: V \to W \), where \( V \) and \( W \) are vector spaces, is the set of all vectors in \( W \) that can be expressed as \( T(v) \) for some \( v \) in \( V \).

Rational sequence topology is a type of topology that can be defined on the set of rational numbers, and it provides a way to study properties of rational numbers using a topological framework. This topology is notably used in mathematical analysis and can be insightful for understanding convergence, continuity, and compactness in contexts where the standard topology on the rationals (induced by the Euclidean topology on the real numbers) may not be ideal.

The Sierpiński carpet is a well-known fractal and two-dimensional geometric figure that exhibits self-similarity. It is constructed by starting with a solid square and recursively removing smaller squares from it according to a specific pattern. Here’s how it is typically created: 1. **Start with a Square**: Begin with a large square, which is often considered a unit square (1 x 1).

The Sorgenfrey plane is a topological space that is constructed from the real numbers, specifically using the Sorgenfrey line as its foundational element. The Sorgenfrey line is obtained by equipping the set of real numbers \(\mathbb{R}\) with a topology generated by half-open intervals of the form \([a, b)\), where \(a < b\). This creates a topology that is finer than the standard topology on \(\mathbb{R}\).

Alejandro Adem is a prominent mathematician known for his contributions to the fields of topology and algebraic geometry. He has worked in areas such as homotopy theory, algebraic topology, and the study of algebraic varieties. Additionally, Adem has held various academic positions and has been involved in research and education in mathematics.

Aleksei Chernavskii might refer to a specific individual, but as of my last knowledge update in October 2023, there isn't widely recognized information or notable references to a person by that name in public sources.

As of my last knowledge update in October 2021, Alexandr Mishchenko is not widely recognized in a specific context such as politics, literature, or entertainment. It's possible that he could be a figure in a specialized field or a more recent individual who gained prominence after my last data cutoff. If you could provide more context or specify the field in which you're interested (e.g.

As of my last update in October 2021, there is no widely recognized figure, concept, or term known as "Charles Newton Little." It's possible that it could refer to a lesser-known individual, a fictional character, or a term that has gained prominence after that date.

Christopher Zeeman is a British mathematician known for his work in topology and the theory of dynamical systems. He is particularly recognized for introducing the concept of "catastrophe theory," which deals with how small changes in parameters can lead to sudden and dramatic shifts in behavior of complex systems. This theory has applications in various fields, including biology, economics, and engineering. Zeeman has also made contributions to the popularization of mathematics and its applications in the real world.

Anatoly Fomenko is a Russian mathematician and historian known for his controversial theories regarding history and chronology. Born on March 13, 1945, Fomenko is a professor at Moscow State University, where he has contributed to various fields, including topology and geometry. Fomenko is best known for his work on "New Chronology," a theory that challenges conventional historical timelines.

Daniel Biss is an American mathematician and politician. He is known for his work in the field of mathematics, particularly in the areas of algebraic geometry and combinatorics. Biss earned a Ph.D. in mathematics from the University of California, Berkeley, and has held academic positions at institutions such as Northwestern University. In addition to his academic career, Biss has also been active in politics.

Daniel Kan is known for being a co-founder of several startups and companies, most notably the ride-sharing service "Fellow," which he co-founded after his time at the well-known startup "Lime" where he worked in various roles. He has also been involved in the tech and entrepreneurial space, contributing to discussions on innovation, business strategies, and technology.