Topology

Topology studies spaces and properties preserved by continuous deformation.

The open sets are the empty set and subsets whose complement is countable. On an uncountable set, any two nonempty open sets intersect, so the space is not Hausdorff. On a countable underlying set this topology is discrete. The compact subsets of a cocountable space are finite, regardless of the cardinality of the underlying set.

Only finite subsets are compact in the cocountable topology. An infinite subset contains distinct $c_1,c_2,\dots$. The open sets obtained by removing the tails $\{c_j:j\ge n\}$ form an increasing open cover, but every finite selection misses a tail point. Finite sets are compact in every topology.

A family $(A_i)$ of subsets of a topological space is locally finite if every point has a neighbourhood meeting only finitely many of its members. On a compact space such a family has only finitely many nonempty members. For singleton subsets of a Riemann surface, this is the condition permitting prescribed poles of a meromorphic function without accumulation inside the surface.

The indiscrete topology on a set $X$ has only the empty set and the whole set as open sets. When $X$ has more than one point it is not a Hausdorff space. For example the quotient topology on $\mathbb{R}/\mathbb{Q}$ is indiscrete: a nonempty open saturated subset contains an interval and all its rational translates, which cover $\mathbb{R}$.

A triangulation of a surface decomposes it into triangles meeting along full edges or vertices. On a closed surface, each edge is incident with two triangle faces, so $3F=2E$. Euler characteristic connects this local counting relation with the surface's topology.

If a space is the union of finitely many closed subspaces and continuous maps on those subspaces agree on every overlap, they combine into a continuous map on the whole space. For a closed target set, its preimage is a finite union of closed sets.

The cofinite topology on a set $X$ consists of the empty set together with every subset whose complement is finite. On an infinite set, any two nonempty open sets intersect.

A nonempty Hausdorff space is the continuous image of $[0,1]$ exactly when it is compact, connected, locally connected, and second-countable.

A Peano continuum is a compact, connected, locally connected metric space. By the Hahn-Mazurkiewicz theorem, every Peano continuum is the image of a continuous surjection from $[0,1]$.

A topological $n$-manifold is a Hausdorff second-countable space in which every point has a neighborhood homeomorphic to an open subset of ${\mathbb{R}}^n$.

Glue two copies of a compact manifold along their boundaries to form its double. A collar gives the double a smooth structure when $W$ is smooth. Folding the copies onto $W$ gives a retraction; in particular, inclusion of either copy is injective on homology. For an oriented manifold, use the opposite orientation on the second copy.

Three intervals joined at one endpoint form a compact connected metric space. It is second countable, Hausdorff and paracompact, but is not a topological manifold. At regular arm points a putative manifold has dimension one; deleting the junction from a small neighborhood instead gives three components, incompatible with an interval chart. This local obstruction precedes any question about a smooth structure.

Invariance of domain says that a continuous map that is an injective function between topological manifolds without boundary of the same dimension is an open map. It is useful in the Penrose singularity theorem: projecting a compact achronal boundary along timelike curves onto a Cauchy hypersurface gives a nonempty subset that is both open and closed, forcing that connected Cauchy hypersurface to be compact.

A smooth manifold carrying a nondegenerate bilinear form on each tangent space, with one negative and all remaining positive eigenvalues, or the overall reversed-sign convention. Its nonzero null vectors have vanishing metric quadratic form.

The metric signature records the signs of the diagonalized nondegenerate metric tensor, equivalently its numbers of negative and positive directions. For a four-dimensional Lorentzian manifold, both $(-,+,+,+)$ and its overall negative convention are common. The numerical signs of contractions and component Hodge star operator identities must be consistent with that choice and the orientation.

In metric signature $(-,+,\dots ,+)$, a spacelike submanifold of a Lorentzian manifold has an induced metric tensor defining a positive-definite quadratic form. With the overall reversed metric signature, its induced metric tensor instead has only negative eigenvalues. The definition applies to any dimension: enclosing charge-integration surfaces in four dimensions are two-dimensional examples, whereas a spacelike Cauchy hypersurface in four dimensions is three-dimensional.

A topological manifold has dimension $n$ when its manifold charts have coordinate domains open in ${\mathbb{R}}^n$. For a smooth manifold, each tangent space has dimension $n$, and a tangent bundle has vector bundle rank $n$.

The codimension is the difference between the dimension of a manifold and that of its submanifold. For complex submanifolds, complex codimension one means real codimension two.

Attaching an index-$k$ handle to an $n$-dimensional manifold with boundary glues $D^k\times D^{n-k}$ along $\partial D^k\times D^{n-k}$. The complementary $D^k\times \partial D^{n-k}$ replaces the attaching region in the boundary.

The core disk of a $k$-handle attachment $D^k\times D^{n-k}$ is $D^k\times \{0\}$. Together with its cap in the preceding stage, it often represents a homology class; for a surgery trace, capped two-handle cores give the basis of its surgery linking matrix.

A handle decomposition builds an $n$-dimensional manifold by successively adjoining $D^h\times D^{n-h}$ along $S^{h-1}\times D^{n-h}$. The attaching data comprise an attaching sphere and its framing of an embedded sphere. Reading a decomposition backwards replaces index $h$ by index $n-h$ and interchanges its attaching spheres and belt spheres.

A handle slide replaces the framed attaching sphere of one handle by a band sum with a parallel copy of another handle of the same index. It changes the attaching data while preserving the diffeomorphism type of the resulting manifold. On the handle incidence matrix it performs an elementary row operation, or a column operation when applied to the dual decomposition.

In the intermediate-index range, if an attaching sphere $A_1$ meets a belt sphere $B_1$ at exactly one transverse point, handle slides can remove the intersections of every other attaching sphere with $B_1$. The dual slides then remove the intersections of $A_1$ with the other belt spheres. Each move uses a band to the unique pivot sheet and a local isotopy removing the old and new intersections. If $m_{ji}=A_j\cdot B_i$ and $\delta =m_{11}=\pm 1$, the remaining incidence block is $m_{ji}-m_{j1}{m}_{1i}/\delta$, the Schur complement of the pivot. Extreme-index cases need separate hypotheses.

The belt sphere of an index-$h$ handle is $\{0\}\times S^{n-h-1}$ in the boundary after attachment. It is the attaching sphere of the dual index-$(n-h)$ handle. Its intersections with the attaching spheres of later handles detect passages through this handle.

The attaching sphere of an index-$h$ handle $D^h\times D^{n-h}$ is $S^{h-1}\times \{0\}$ in the previous boundary. Its framing of an embedded sphere identifies a neighbourhood with $S^{h-1}\times D^{n-h}$ and specifies the handle attachment.

A three-manifold is a topological manifold locally modeled on ${\mathbb{R}}^3$, or on a half-space at its boundary.

A compact three-manifold is boundary-irreducible when there is no embedded disk with interior in the manifold and boundary an essential curve in its boundary. Thus its boundary has no compressing disk.

An orientable three-manifold is irreducible if every embedded two-sphere bounds a three-ball. This makes innermost-disk simplifications of embedded topological surfaces possible without changing the ambient three-manifold.

In the orientable case an excellent compact three-manifold is an irreducible three-manifold and a boundary-irreducible three-manifold, is not a ball, contains a two-sided properly embedded incompressible surface, and has every properly embedded incompressible zero-Euler characteristic surface a boundary-parallel surface. In particular it has no essential annulus or torus. These properties give lower bounds on the Thurston norm of classes with essential boundary.

In a compact connected orientable three-manifold with no spherical boundary components, every embedded knot is homotopic to a knot with an excellent three-manifold as exterior. In particular this can preserve the generator class in $S^1\times S^2$, or the winding number of a satellite pattern one class in a solid torus. Homotopy here is weaker than isotopy.

For a compact oriented three-manifold, the integral relative class $u$ has norm the minimum of ${\chi }_{-}(F)={\sum }_j\max (0,-\chi (F_j))$ over properly embedded oriented topological surfaces representing $u$. Homogeneity and continuity extend it to real relative homology. It is a seminorm, since spheres, disks, annuli and tori have zero cost.

For $Y=P\times S^1$ with $P$ a pair of pants and $h$ its circle fiber, a horizontal surface in a Seifert fibered space of degree $d$ has ${\chi }_{-}=|d|$; vertical surfaces in a Seifert fibered space have zero cost. Cutting and pasting these representatives, and classifying essential surfaces, proves $\Vert u{\Vert }_T=|u\cdot h|$. The dual Thurston polytope is therefore the segment between the two functionals $\pm h$ under the natural pairing.

In an irreducible three-manifold with incompressible boundary, a minimizing surface can be cut along a separating incompressible torus after removing inessential intersection circles. The essential pieces have no disk or sphere components, so their ${\chi }_{-}$ costs add. Minimizing independently in the pieces and gluing their matching boundary curves gives the corresponding upper bound. This allows Thurston norm calculations for satellite knots.

The dual polytope consists of the real linear functionals $\alpha$ on $H_2(Y,\partial Y;\mathbb{R})$ such that $|\alpha (u)|\le \Vert u{\Vert }_T$ for every $u$. Thus it is the polar of the unit ball of the Thurston norm and annihilates that seminorm's kernel.

An incompressible surface is a properly embedded two-sided topological surface with no compressing disk, subject to the usual exclusion of inessential spherical components. A compressing disk has boundary an essential closed curve on the topological surface and interior disjoint from it.

A properly embedded topological surface is boundary-parallel if it cobounds a product region with a topological surface in the ambient boundary, with the remaining boundary of that product contained in the ambient boundary. An annulus parallel into one boundary component has zero net oriented boundary class on that component.

In an orientable irreducible Seifert fibered space, an essential two-sided incompressible surface that is also a boundary-incompressible surface can be isotoped to a horizontal surface in a Seifert fibered space or a vertical surface in a Seifert fibered space.

A properly embedded topological surface is boundary-incompressible when no disk in the ambient three-manifold joins an essential arc on the topological surface to an arc on the ambient boundary, with the disk's interior disjoint from both.

A Heegaard splitting divides a closed three-manifold into two handlebodies with common boundary. For a three-manifold with boundary, one uses compression bodies, which can retain negative boundary components.

A compression body is obtained from a product $F\times I$ by attaching one-handles to $F\times \{1\}$, allowing three-ball components as well. Its negative boundary is the retained copy of $F$ and its positive boundary is the other boundary component. A handlebody has empty negative boundary.

The Heegaard surface is the common positive boundary of the two pieces in a Heegaard splitting.

A Heegaard diagram marks the boundaries of compressing disks from the two sides of a Heegaard surface. Attaching two-handles along the two collections, and capping resulting spherical boundary components, reconstructs the three-manifold.

A generalized Heegaard diagram permits compression bodies and a nonempty negative boundary. Equivalently, a suitable handlebody with two-handles attached along only part of a compressing system can retain torus boundary components. Disk crossings supply generators and attaching-curve words supply relators for its fundamental group.

Thicken the planar graph underlying a knot diagram. Its boundary supports a Heegaard diagram with face disks on one side and crossing-tunnel disks on the other. The face generators and crossing curves give a Dehn presentation of a knot group.

A genus-$g$ handlebody is a three-ball with $g$ one-handles attached. Its boundary is a closed orientable topological surface of genus $g$.

Every connected closed orientable three-manifold with finite fundamental group admits spherical geometry. With cyclic fundamental group, it is a lens space.

An $n$-manifold has zero singular homology in every degree greater than $n$. One proof uses a countable exhaustion by subsets that retract onto $n$-dimensional CW complexes and the compact support of every singular cycle.

A closed set contains all its limit points; equivalently, its complement is open.

A sequentially closed set is one that contains every limit of its convergent sequences. This is weaker than being a closed set in a general topological space, but the two conditions agree in metric spaces.

A limit point of a set is a point whose every neighbourhood contains a different point of the set.

The closure of a subset is the smallest closed set containing it, equivalently the set of points whose every neighbourhood meets it.

An adherent point of $A$ is a point whose every neighbourhood meets $A$. The set of adherent points is the closure of $A$. This includes all points of $A$, even isolated ones; a limit point in the accumulation-point convention instead requires each neighbourhood to meet $A\setminus \{x\}$.

The interior of a subset $A$ of a topological space is the union of all open sets contained in $A$, equivalently the largest open subset of $A$.

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.

For an equivalence relation $R$ on a topological space $X$, the quotient topology on the set $X/R$ declares $U\subseteq X/R$ open exactly when its inverse image under the projection $q:X\to X/R$ is open in $X$.

A quotient topological space identifies points of a topological space according to an equivalence relation and equips the set of equivalence classes with the quotient topology. A subset is open exactly when its inverse image under the canonical quotient map is open. Even a Hausdorff space can have a non-Hausdorff quotient; identifying real numbers differing by a rational number gives a multi-point quotient with the indiscrete topology.

A quotient map is a surjective continuous map $q:X\to Y$ for which $U\subseteq Y$ is open exactly when $q^{-1}(U)$ is open.