Mathematics / Topology / Point-set topology

Topology notes

Quotient Spaces Topological Spaces and Point-Set Topology

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Working Intuition: Topology Without Distance

The quickest way into topology is not to memorize the three axioms first. It is to ask what part of analysis survives after one removes the numerical distance function.

In a metric space, proximity is measured by balls

B_r(x)=\{y\in X:d(x,y)<r\}.

Limits, continuity, neighborhoods, and local arguments are all phrased using these balls. But many geometric constructions do not really need the number $d(x,y)$ itself. They only need to know which subsets count as neighborhoods, which properties can be checked locally, and which points cannot be separated by the available open sets.

A topology keeps exactly that information. It forgets the ruler and remembers the pattern of open sets.

This is why the axioms for open sets are not arbitrary. In metric spaces:

the empty set and the whole space are open;
an arbitrary union of open sets is open, because being locally inside one of them is enough;
a finite intersection of open sets is open, because finitely many local conditions can be satisfied at once by shrinking the radius.

The finite-intersection condition is important. If one tries to satisfy infinitely many radius constraints at once, the allowed radius can collapse to zero. Topology is built to preserve the kind of local reasoning that actually works.

Neighborhoods Are the Real Primitive

Although one defines a topology as a collection of open subsets, the mental object is often the neighborhood system of each point. A neighborhood of $x$ is something large enough to contain an open set around $x$ . Once neighborhoods are known, we can define:

convergence: eventually every neighborhood is entered;
limit points: every neighborhood meets the set again;
closure: all points forced by neighborhoods;
continuity: inverse images preserve neighborhoods/open sets;
local structure: a property is checked after restricting to neighborhoods.

This is the bridge from analysis to geometry. A manifold is locally Euclidean because every point has a neighborhood that looks like an open subset of $\mathbb R^n$ . A fibre bundle is locally a product because every base point has a neighborhood over which the total space looks like $U\times F$ . In both cases, topology tells us what “locally” means before calculus or linear algebra enters.

Closed Sets, Closure, and the Common Confusion

Closed does not mean “not open.” A set is closed if its complement is open. A set can be both open and closed, or neither.

The useful intuition is:

\overline A=\text{the set }A\text{ together with all points that }A\text{ cannot avoid locally.}

Equivalently, $x\in\overline A$ if every neighborhood of $x$ intersects $A$ . A closed set is one that already contains all such forced boundary or limiting points.

The interior goes in the opposite direction:

\operatorname{int}(A)=\text{the part of }A\text{ that is safely surrounded by }A.

So closure asks what must be added to make $A$ topologically complete from the outside; interior asks what part of $A$ is robustly inside.

Continuity as Pulling Back Observations

The definition

f:X\to Y \text{ is continuous}\quad\Longleftrightarrow\quad f^{-1}(U)\text{ is open in }X\text{ for every open }U\subset Y

is one of the cleanest ideas in point-set topology.

It says: every observable open condition in the target can be checked by an open condition in the source. We use preimages rather than images because preimages preserve unions, finite intersections, the empty set, and the whole set automatically. Images do not.

A homeomorphism is then a change of description that loses no topological information. Two spaces are homeomorphic when they have the same open-set structure, even if they are embedded, drawn, or parametrized differently.

Why This Matters Later

For the notes that follow, topology supplies the grammar for gluing.

Quotient spaces describe what happens when points are identified.
Product and subspace topologies describe how local coordinate pieces sit inside larger constructions.
Hausdorffness prevents points from being topologically indistinguishable.
Compactness turns local covers into finite data.
Paracompactness and partitions of unity let local constructions be patched globally.

The fibre-bundle language later in these notes is built on this foundation: local triviality is a statement about open neighborhoods, transition functions live on overlaps, and total spaces are often constructed by quotienting a disjoint union of local pieces.

Point-Set Topology Refresher

Purpose of Appendix D This appendix reviews the point-set topology needed to read manifold and bundle definitions without ambiguity. It assumes only elementary set theory. The emphasis is on definitions and theorems that explain phrases such as “locally homeomorphic”, “open cover”, “compact”, “Hausdorff”, “quotient topology”, “product topology”, and “paracompact”.

Topological spaces

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

$\emptyset\in\mathcal T$ and $X\in\mathcal T$,
arbitrary unions of elements of $\mathcal T$ are in $\mathcal T$,
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:

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

Euclidean topology The usual topology on $\mathbb R^n$ has basis the open balls \[B_r(x)=\{y\in\mathbb R^n:|y-x|<r\}.\] It also has basis the open rectangles \[(a_1,b_1)\times\cdots\times(a_n,b_n).\]

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.

Why quotient topology matters When a bundle is formed by gluing pieces $U_i\times F$ using transition functions, the total space is often constructed as a quotient of a disjoint union. The topology on the quotient is the quotient topology.

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.

Theorem 12 (Urysohn lemma, statement). If $X$ is normal and $A,B\subset X$ are disjoint closed sets, then there exists a continuous function \[f:X\to[0,1]\] such that $f|_A=0$ and $f|_B=1$.

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.

Theorem 13 (Heine-Borel theorem). A subset of $\mathbb R^n$ is compact in the Euclidean topology if and only if it is closed and bounded.

Theorem 14 (Continuous image of compact is compact). If $f:X\to Y$ is continuous and $X$ is compact, then $f(X)$ is compact.

Theorem 15 (Compact to Hausdorff). If $X$ is compact, $Y$ is Hausdorff, and $f:X\to Y$ is a continuous bijection, then $f$ is a homeomorphism.

Compactness is not part of the definition of a fibre 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:

$\operatorname{supp}\rho_i\subset U_i$,
the supports are locally finite,
$\sum_i\rho_i(x)=1$ 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 fibre bundles with discrete fibre. 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:

$M$ is Hausdorff,
$M$ is second-countable,
every point has a neighborhood homeomorphic to an open subset of $\mathbb R^n$.

A smooth manifold adds a maximal smooth atlas.

A topological fibre bundle $\pi:E\to B$ with fibre $F$ 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

Theorem 16 (Tychonoff theorem). An arbitrary product of compact topological spaces is compact in the product topology.

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.

Theorem 17 (Baire category theorem). A complete metric space is not a countable union of nowhere dense closed sets. More generally, locally compact Hausdorff spaces are Baire spaces.

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

Tietze extension theorem

Theorem 18 (Tietze extension theorem). If $X$ is normal, $A\subset X$ is closed, and $f:A\to\mathbb R$ is continuous and bounded, then $f$ extends to a continuous function \[F:X\to\mathbb R.\]

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.

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-fibre prototype of fibre bundles.

Jonginn Yun

Topological Spaces and Point-Set Topology