Topological space

A topological space is a set together with a family of open subsets containing the empty set and the whole set and closed under arbitrary unions and finite intersections.

A topological space is separable when it has a countable dense subset. The topology is part of the assertion: a dual may be separable in its weak-star topology and not in its norm topology. A compact metric space is separable by taking the union of finite nets at scales tending to zero.

A G-delta set is a countable intersection of open subsets of a topological space. In a complete metric space these are precisely the topologically complete subspaces, by the G-delta criterion for topological completeness.

A net is a family indexed by a directed ordered set: every pair of indices has a common upper bound. It converges to $x$ when it is eventually in every neighbourhood of $x$. Unlike sequences, nets characterize closure in every topological space: a point is in the closure of a set precisely when some net in that set converges to it.

A subnet is obtained from a net by an order-preserving cofinal map from another directed index set. It preserves any limit of the original net, and eventual properties still hold along it. Every net in a compact set has a convergent subnet. This supplies compactness arguments when sequential compactness is unavailable.

A separation axiom is a condition on a topological space expressing how points or closed subsets can be distinguished by open sets. The Kolmogorov space condition distinguishes distinct points by some open set; the T1 space condition makes every point closed; the stronger Hausdorff space condition gives disjoint neighbourhoods to distinct points.

A topological space is T1 when every point is a closed point. Equivalently, for distinct $x,y$ there is an open set containing $x$ but not $y$, and an open set containing $y$ but not $x$. These open sets need not be disjoint. Every Hausdorff space is T1, but an infinite set with its cofinite topology is T1 without being Hausdorff.

A point of a topological space is closed when its singleton is a closed set. In a T1 space every point is closed. The closed points of an affine scheme $\mathrm{Spec}A$ are its maximal ideals; the closure of the point corresponding to $\mathfrak{p}$ is $V(\mathfrak{p})$.

A topological space has local connectedness when every point has a neighborhood basis of connected open sets. In particular, a planar domain has local connectedness, but its domain boundary need not be. Local connectedness of the boundary is the condition governing a continuous extension of a conformal bijection to the parametrizing circle.

The boundary of a subset $A$ of a topological space consists of the points whose every neighborhood meets both $A$ and its complement. Equivalently, $\partial A=\overline{A}\setminus A^{\circ }$, where the bar denotes closure and the circle denotes interior.

The boundary of a planar domain $D$ is its boundary of a set. It is distinct from the boundary of a parametrizing interval: a curve can begin on $\partial D$ while all its positive-time points lie inside $D$.

A basis of a topology is a family of open sets such that every open set is a union of members of that family. Equivalently, every point of every open set lies in a basis member contained in it. The rectangles $U\times V$ form a basis for the product topology of two spaces. This topological meaning differs from a linear-algebra basis.

A second-countable topological space has a countable collection of open sets such that every open set is a union of members of that collection. A space covered by countably many open manifold charts is second countable because each coordinate domain has a countable basis of rational balls.

A topological space is paracompact when every open cover has a locally finite open refinement. Every metric space and every second-countable topological manifold is paracompact.

A topological space is $T_0$ when every pair of distinct points is distinguished by an open set containing exactly one of them.

The Sierpiński space is $\{0,1\}$ with open sets $\varnothing$, $\{1\}$, and $\{0,1\}$. Continuous maps $X\to S$ are in bijection with open subsets of $X$ through $g\mapsto g^{-1}(1)$.

For a T0 space $X$, evaluation against all continuous maps $g:X\to S$ is a homeomorphism onto its image under the mapinto a power of the Sierpiński space. The $T_0$ axiom makes it injective, and the coordinate inverse images of $\{1\}$ recover every open subset of $X$.

A map between topological spaces is continuous when the inverse image of every open set is open.

A continuous map injective map $f:X\to Y$ that is a homeomorphism onto its image with the subspace topology. A continuous map injective map from a compact space into a Hausdorff space is a topological embedding.

A retraction of a topological space $X$ onto a subspace $A$ is a continuous map $r:X\to A$ whose restriction to $A$ is the identity.

A once-punctured genus-$h$ closed orientable surface retracts to a wedge of $2h$ circles. A retraction from ${\Sigma }_g$ would inject its degree-one cohomology as an isotropic subspace of a symplectic vector space for the Poincare duality pairing, since the punctured surface has zero degree-two cohomology. The maximal isotropic dimension is $g$, giving $2h\le g$. Equality is realized by folding the double of the punctured surface onto one copy. Additional handles in the other copy can be pinched away while fixing the joining boundary, proving sufficiency for every embedding of such a one-boundary subsurface when $g\ge 2h$.

For $A\subseteq X$, the subspace topology consists of the intersections $A\cap U$ with open subsets $U\subseteq X$.

The one-point compactification of a locally compact noncompact Hausdorff space adjoins one point whose neighbourhoods are complements of compact subsets. The result is compact Hausdorff.

The Hawaiian earring is the union of a countable family of circles tangent at one point, with their radii tending to zero, endowed with its planar subspace topology. Equivalently it is the one-point compactification of a countable disjoint union of real lines. Every neighbourhood of the common point contains all but finitely many whole circles. This compact topological space is not locally contractible there and has different singular homology from the infinite CW complex wedge of circles.

The first integral singular homology of the Hawaiian earring has a direct summand isomorphic to $\mathbb{Q}$. This consequence of its structural homology theorem suffices, through the universal coefficient theorem for cohomology and nonzero Ext of the rationals with integer coefficients, to show that its integral $H^2$ is nonzero. The precise structural theorem is Theorem 3.1 of www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/eda-kawamura2.pdf.

The circle is the subspace $S^1=\{z\in \mathbb{C}:|z|=1\}$, equivalently the quotient of a closed interval obtained by identifying its endpoints.

A cyclic chain of distinct circles between two disjoint circle boundaries, each tangent to both boundaries and its two neighbours. In concentric coordinates with inner radius $\rho$ and outer radius $R$, the chain circles have radius $r=(R-\rho )/2$ and centre distance $d=(R+\rho )/2$. A simple disjoint chain with $n\ge 3$ closes when $r/d=\sin (\pi /n)$.

If a simple Steiner chain closes after $n\ge 3$ circles, rotating the starting circle between concentric boundaries gives another closed $n$-circle chain. Applying a Möbius transformation proves the corresponding statement for arbitrary boundary circles. Construction must stay in the same intervening family and proceed without reversing the chosen direction. Merely choosing arbitrary circles tangent to the preceding circle permits backtracking and does not assert closure.

For a Steiner chain between concentric circles, tangencies of neighbouring circles are midpoints of their centres and lie on the circle of radius $\sqrt{R\rho }$. Under a Möbius transformation, their locus is a generalized circle, which may be a line. An ordinary-circle conclusion requires the pole of the inverse transformation to avoid the original locus.

An equation $|az+b\overline{z}|=ux+v$, with $z=x+iy$ and real coefficients, becomes $(a+b{)}^2{x}^2+(a-b{)}^2{y}^2=(ux+v{)}^2$ after squaring. If $(a+b{)}^2-u^2=(a-b{)}^2>0$, completing the square gives a circle. The original locus is only the part where $ux+v\ge 0$. Checking this sign on the candidate circle prevents extraneous points introduced by squaring. This method combines the complex modulus with elementary conic geometry.

The radius of a circle or sphere is the Euclidean distance from its centre to its boundary. The diameter is twice the radius.

Pi is the ratio of a Euclidean circle's circumference to its diameter.

Assuming $\pi =a/b$ and integrating a suitably normalized polynomial times $\sin x$ produces a positive integer that tends to zero as the polynomial degree grows. This contradiction proves that $\pi$ is an irrational number.

The wedge sum of pointed spaces identifies all their basepoints. In particular, a wedge of $m$ circles has first homology ${\mathbb{Z}}^m$.

A pinch map collapses a separating subspace to a point so that the quotient splits into a wedge sum. For a connected sum of oriented manifolds, collapsing the separating boundary sphere gives a map $M\#N\to M\vee N$. The two wedge projections have degree of a continuous mapping one with compatible orientations. Consequently their pullbacks identify the two top orientation classes, while positive-degree classes from different summands have zero cup product.

The wedge of two circles is a graph with one vertex and two loop edges. Its fundamental group is the free group on the two oriented loops.

For a subspace $A\subseteq X$, the inclusion map is the injective function $i:A\to X$ given by $i(a)=a$.

A family $\mathcal{T}$ of subsets of a set $X$ is a topology when $\varnothing ,X\in \mathcal{T}$, every union of members of $\mathcal{T}$ belongs to $\mathcal{T}$, and every finite intersection of members of $\mathcal{T}$ belongs to $\mathcal{T}$.

An open cover of a subset $A$ is a family of open sets whose union contains $A$.

An open cover is acyclic for a specified sheaf if its restriction to every nonempty finite intersection has zero higher sheaf cohomology. The acyclic cover theorem then identifies the cover's Čech cohomology with sheaf cohomology. On a separated scheme, an affine open cover is acyclic for every quasi-coherent sheaf.

A topological space is quasi-compact when every open cover has a finite subcover. This is the form of compactness used for spaces that need not be Hausdorff, especially spectra and schemes.