Point-Set Topology Refresher

Topological spaces

A topology on a set $X$ is a collection $\mathcal T$ of subsets of $X$, called open sets, such that:

1. $\emptyset\in\mathcal T$ and $X\in\mathcal T$,

2. arbitrary unions of elements of $\mathcal T$ are in $\mathcal T$,

3. finite intersections of elements of $\mathcal T$ are in $\mathcal T$.

The pair $(X,\mathcal T)$ is a topological space.

A subset $C\subset X$ is closed if $X\setminus C$ is open. A set can be both open and closed, or neither.

Bases and subbases

A basis for a topology on $X$ is a collection $\mathcal B$ of subsets of $X$ such that:

1. for every $x\in X$, there exists $B\in\mathcal B$ with $x\in B$,

2. if $x\in B_1\cap B_2$ with $B_1,B_2\in\mathcal B$, then there exists $B_3\in\mathcal B$ such that

$$x\in B_3\subset B_1\cap B_2.$$

The topology generated by $\mathcal B$ consists of arbitrary unions of basis elements.

A subbasis $\mathcal S$ is a collection of subsets whose finite intersections form a basis.

Continuity

A map $f:X\to Y$ between topological spaces is continuous if

$$f^{-1}(U)\subset X$$

is open for every open set $U\subset Y$.

This definition is intentionally formulated using preimages, not images. Images of open sets under continuous maps need not be open.

A homeomorphism is a bijection $f:X\to Y$ such that both $f$ and $f^{-1}$ are continuous. If such a map exists, $X$ and $Y$ are topologically the same.

Initial, final, product, and subspace topologies

The subspace topology on $A\subset X$ is

$$\mathcal T_A=\{A\cap U:U\text{ open in }X\}.$$

The product topology on $X\times Y$ is generated by basis sets

$$U\times V, \qquad U\subset X\text{ open},\quad V\subset Y\text{ open}.$$

For an arbitrary product $\prod_{i\in I}X_i$, the product topology is generated by finite restrictions: basis elements restrict only finitely many coordinates and leave the rest unrestricted.

The quotient topology is defined as follows. Let $q:X\to Y$ be a surjective map. A subset $U\subset Y$ is declared open if and only if

$$q^{-1}(U)$$

is open in $X$. This is the finest topology on $Y$ making $q$ continuous.

Closure, interior, boundary, and dense subsets

The closure of $A\subset X$ is

$$\overline A=\bigcap\{C\subset X:C\text{ closed and }A\subset C\}.$$

The interior is

$$\operatorname{int}(A)=\bigcup\{U\subset A:U\text{ open}\}.$$

The boundary is

$$\partial A=\overline A\setminus\operatorname{int}(A).$$

A subset $A$ is dense in $X$ if

$$\overline A=X.$$

Equivalently, every nonempty open set intersects $A$.

Neighborhoods and local properties

A neighborhood of $x\in X$ is a subset $N\subset X$ containing an open set $U$ with $x\in U\subset N$. A property is local if it can be checked on some neighborhood of each point.

A space $X$ is locally Euclidean of dimension $n$ if for every $x\in X$ there exists an open neighborhood $U\subset X$ and a homeomorphism

$$\varphi:U\to V$$

onto an open subset $V\subset\mathbb R^n$. A Hausdorff, second-countable, locally Euclidean space is a topological manifold.

Separation axioms

A space $X$ is $T_0$ if any two distinct points are topologically distinguishable. It is $T_1$ if singletons are closed. It is Hausdorff, or $T_2$, if for any distinct points $x,y\in X$ there exist disjoint open sets $U,V$ such that

$$x\in U, \qquad y\in V.$$

Hausdorffness guarantees uniqueness of limits of sequences in first-countable spaces and is part of the standard definition of manifolds.

A space is regular if points and closed sets can be separated by neighborhoods. It is normal if disjoint closed sets can be separated by neighborhoods. Normal spaces support powerful extension theorems.

Countability axioms

A space is first-countable if every point has a countable neighborhood basis. Metric spaces are first-countable.

A space is second-countable if its topology has a countable basis. Smooth manifolds are usually required to be second-countable. This excludes pathological disjoint unions with too many components and ensures many analysis tools behave well.

A space is separable if it has a countable dense subset. In metric spaces, second-countability implies separability, and separability often implies second-countability under additional hypotheses.

Compactness

An open cover of $X$ is a collection $\{U_i\}_{i\in I}$ of open subsets such that

$$X=\bigcup_{i\in I}U_i.$$

A subcover is a subcollection that still covers $X$. The space $X$ is compact if every open cover has a finite subcover.

Compactness is not part of the definition of a fiber bundle or a manifold. It is an extra global finiteness property. Many base spaces in physics, such as the Brillouin torus $T^d$, are compact; spacetime manifolds often are not.

Local compactness and one-point compactification

A space is locally compact if every point has a neighborhood whose closure is compact, or equivalently in Hausdorff spaces, every point has a compact neighborhood.

If $X$ is locally compact, Hausdorff, and noncompact, its one-point compactification is

$$X^+=X\cup\{\infty\}$$

with open sets consisting of open subsets of $X$ and sets containing $\infty$ whose complements in $X$ are compact closed subsets.

Example:

$$\mathbb R^n\cup\{\infty\}\cong S^n.$$

Connectedness and path connectedness

A separation of $X$ is a decomposition

$$X=U\cup V$$

where $U,V$ are nonempty disjoint open subsets. A space is connected if it has no separation.

A path in $X$ is a continuous map

$$\gamma:[0,1]\to X.$$

A space is path connected if any two points can be joined by a path. Path connected implies connected, but connected need not imply path connected.

Connected components are maximal connected subsets. Path components are maximal path connected subsets.

Metric spaces as topological spaces

A metric on $X$ is a function

$$d:X\times X\to[0,\infty)$$

satisfying positivity, symmetry, and triangle inequality. It induces a topology whose basis is open balls.

A sequence $x_n$ converges to $x$ if

$$\lim_{n\to\infty}d(x_n,x)=0.$$

A sequence is Cauchy if for every $\epsilon>0$ there exists $N$ such that

$$m,n\ge N\Rightarrow d(x_m,x_n)<\epsilon.$$

A metric space is complete if every Cauchy sequence converges.

Nets and why sequences are not always enough

In general topological spaces, sequences may fail to detect closure. A net is a generalized sequence indexed by a directed set. A directed set is a set $I$ with a preorder such that any two elements have a common upper bound. A net in $X$ is a map

$$I\to X, \qquad i\mapsto x_i.$$

A net converges to $x$ if for every neighborhood $U$ of $x$, there exists $i_0\in I$ such that $x_i\in U$ whenever $i\ge i_0$.

The topological characterization is:

$$x\in\overline A \quad\Longleftrightarrow\quad \text{there is a net in }A\text{ converging to }x.$$

In first-countable spaces, sequences suffice.

Paracompactness and partitions of unity

An open cover $\mathcal V$ refines an open cover $\mathcal U$ if every set in $\mathcal V$ is contained in some set in $\mathcal U$. A cover is locally finite if every point has a neighborhood intersecting only finitely many sets in the cover.

A Hausdorff space is paracompact if every open cover has a locally finite open refinement. Smooth manifolds that are Hausdorff and second-countable are paracompact.

A partition of unity subordinate to an open cover $\{U_i\}$ is a collection of continuous or smooth functions $\{\rho_i:X\to[0,1]\}$ such that:

1. $\operatorname{supp}\rho_i\subset U_i$,

2. the supports are locally finite,

3. $\sum_i\rho_i(x)=1$ for every $x\in X$.

Partitions of unity allow local data to be patched into global data. They are the reason vector bundles over paracompact manifolds admit connections.

Topological groups

A topological group is a group $G$ with a topology such that multiplication and inversion are continuous:

$$G\times G\to G, \qquad (g,h)\mapsto gh,$$ $$G\to G, \qquad g\mapsto g^{-1}.$$

A Lie group is a topological group with a smooth manifold structure for which multiplication and inversion are smooth.

Covering spaces

A covering map is a continuous surjection

$$p:\widetilde X\to X$$

such that every $x\in X$ has an open neighborhood $U$ with

$$p^{-1}(U)=\bigsqcup_{\alpha\in A}V_\alpha$$

where each restriction

$$p|_{V_\alpha}:V_\alpha\to U$$

is a homeomorphism.

Covering spaces are fiber bundles with discrete fiber. The universal covering group $SU(2)\to SO(3)$ is both a covering map and a Lie group homomorphism.

Point-set checklist for manifolds and bundles

A topological manifold of dimension $n$ is usually defined as a topological space $M$ satisfying:

1. $M$ is Hausdorff,

2. $M$ is second-countable,

3. every point has a neighborhood homeomorphic to an open subset of $\mathbb R^n$.

A smooth manifold adds a maximal smooth atlas.

A topological fiber bundle $\pi:E\to B$ with fiber $F$ requires an open cover $\{U_i\}$ of $B$ such that

$$\pi^{-1}(U_i)\cong U_i\times F$$

over $U_i$. The condition is that at least one such trivializing cover exists. It is not required that every open cover trivialize the bundle.

Interior, closure, and continuity by closures

For $A\subset X$, one has the useful identities

$$x\in\overline A \quad\Longleftrightarrow\quad \text{every neighborhood of }x\text{ intersects }A.$$

A map $f:X\to Y$ is continuous if and only if

$$f(\overline A)\subset\overline{f(A)}$$

for every $A\subset X$. This formulation often helps when reasoning about limits and dense subsets.

Open maps, closed maps, and quotient maps

A map $f:X\to Y$ is open if it sends open sets to open sets. It is closed if it sends closed sets to closed sets. A quotient map is a surjective map $q:X\to Y$ such that $U\subset Y$ is open exactly when $q^{-1}(U)$ is open.

Every quotient map is continuous by definition, but not every continuous surjection is a quotient map. Open continuous surjections and closed continuous surjections are quotient maps.

This matters for bundles because local trivializations give homeomorphisms locally, but quotient constructions must still be given the correct topology globally.

Tychonoff theorem and product compactness

For finite products this is elementary compared to the general theorem. Infinite products require the product topology, not the box topology. The theorem is equivalent to the axiom of choice in standard set theory.

Box topology versus product topology

For a product $\prod_{i\in I}X_i$, the box topology has basis sets

$$\prod_{i\in I}U_i$$

with every $U_i\subset X_i$ open. The product topology allows only finitely many coordinates to be restricted at a time. For infinite products, the box topology is usually too fine and fails to preserve compactness.

Compact-open topology

For spaces $X,Y$, the compact-open topology on the function space $C(X,Y)$ is generated by subbasis sets

$$[K,U]=\{f\in C(X,Y):f(K)\subset U\},$$

where $K\subset X$ is compact and $U\subset Y$ is open. This topology is important when studying homotopies, loop spaces, and mapping spaces.

Baire category theorem

A subset of a topological space is nowhere dense if the interior of its closure is empty. A set is meagre if it is a countable union of nowhere dense sets.

This theorem explains why many “generic” properties in analysis and geometry are expressed as countable intersections of open dense sets.

Tietze extension theorem

This theorem is a close relative of Urysohn’s lemma and helps explain why normality is a strong separation property.

Metrization ideas

A space is metrizable if its topology comes from a metric. Not every topological space is metrizable. Metrization theorems give conditions under which a topology is induced by a metric.

One standard result is that every second-countable regular Hausdorff space is metrizable. Since smooth manifolds are Hausdorff and second-countable, and are locally Euclidean, they are metrizable as topological spaces. The smooth structure, however, is additional information not determined by an arbitrary metric inducing the topology.

Locally finite covers and refinement

A cover $\mathcal V$ refines a cover $\mathcal U$ if every $V\in\mathcal V$ lies in some $U\in\mathcal U$. A star refinement is a stronger refinement useful in metrization and paracompactness arguments.

Paracompactness is important because it upgrades local constructions to global constructions. In differential geometry, this is what allows one to choose Riemannian metrics and connections on arbitrary smooth manifolds satisfying the usual hypotheses.

Topological manifolds revisited

A chart on a topological manifold $M$ is a pair $(U,\varphi)$ where $U\subset M$ is open and

$$\varphi:U\to \varphi(U)\subset\mathbb R^n$$

is a homeomorphism onto an open subset. If $(U,\varphi)$ and $(V,\psi)$ are charts with $U\cap V\ne\emptyset$, the transition map is

$$\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V).$$

A smooth atlas requires these transition maps to be smooth. A topological manifold does not require smoothness.

CW complexes

A CW complex is built inductively by attaching cells. Start with a discrete set of points $X^0$. Attach $n$-cells by maps

$$S^{n-1}\to X^{n-1}$$

and form pushouts. The resulting space is

$$X=\bigcup_{n\ge0}X^n.$$

CW complexes are topologically manageable and are the natural setting for many results in algebraic topology. Many spaces in physics, including spheres, tori, and projective spaces, admit CW structures.

Proper maps

A continuous map $f:X\to Y$ is proper if the preimage of every compact set is compact:

$$K\subset Y\text{ compact}\Rightarrow f^{-1}(K)\subset X\text{ compact}.$$

For locally compact Hausdorff spaces, proper maps behave like maps that preserve behavior at infinity. Properness is often the right replacement for compactness of the domain.

Topological summary for bundles

The topological inputs needed for bundles are:

• open covers to formulate local triviality,

• product topology for $U\times F$,

• quotient topology for gluing local pieces,

• Hausdorff and second-countability assumptions for manifolds,

• paracompactness for partitions of unity and existence of connections,

• covering spaces as the discrete-fiber prototype of fiber bundles.