A Crash Course in Topology

This is a tour of the landmarks beyond Topological Spaces and Metric Spaces. Order of climb: first see how topologies are generated, then ascend the axiom ladder (separation, countability, compactness), then meet the invariants (algebraic topology, Euler characteristic), and finally connect back to fixed-point arguments — the bridge to game theory and mechanism design.

Generating Topologies

Listing all open sets is intractable. We generate them.

Basis

A basis $\mathcal{B}$ for a topology on $X$ is a family of subsets satisfying:

  1. $\forall x \in X,\ \exists B \in \mathcal{B} : x \in B$
  2. If $x \in B_1 \cap B_2$, $\exists B_3 \in \mathcal{B} : x \in B_3 \subseteq B_1 \cap B_2$.

Open sets are then arbitrary unions of basis elements. Equivalently, $U$ is open iff $\forall x \in U,\ \exists B \in \mathcal{B} : x \in B \subseteq U$.

[!note] Generating analogy A basis is to a topology what a generating set is to a vector space, or productions are to a grammar — a small description closed under prescribed operations (here: arbitrary union, finite intersection).

The canonical basis on a metric space is $\{B(x, r)\}_{x, r>0}$. This is exactly the route that takes Metric Spaces to topologies.

Subbasis

A subbasis $\mathcal{S}$ is any family covering $X$. The induced basis is the set of all finite intersections of subbasis elements; the topology is then arbitrary unions of those.

[!tip] Why this matters The product topology on $\prod_{i \in I} X_i$ is defined as the topology generated by the subbasis $\{\pi_i^{-1}(U_i) : U_i \text{ open in } X_i\}$. This is the coarsest topology making every projection continuous — a universal property.

Subspace topology

For $A \subseteq X$: $\mathcal{T}_A = \{A \cap U : U \in \mathcal{T}\}$. The smallest topology on $A$ making the inclusion $A \hookrightarrow X$ continuous. Warning: open in $A$ ≠ open in $X$; e.g. $[0, \tfrac{1}{2})$ is open in $[0,1]$ but not in $\mathbb{R}$.

Product topology (revisited)

Finite products: $U \times V$ is a basis. Infinite products $\prod_{i \in I} X_i$ have the crucial restriction that basic opens look like $\prod_i U_i$ where $U_i = X_i$ for all but finitely many $i$. The naive “box topology” (any product of opens) is strictly finer and kills compactness theorems.

Quotient topology

Given $X$ and $\sim$, let $q: X \to X/\!\sim$ be the projection. Then $U \subseteq X/\!\sim$ is open iff $q^{-1}(U)$ is open in $X$. The universal property: a map $\bar f : X/\!\sim \to Y$ is continuous iff $\bar f \circ q : X \to Y$ is. This is the workhorse for building non-trivial spaces.

Construction Identification Result
Square $[0,1]^2$ $(0,y) \sim (1,y)$ and $(x,0) \sim (x,1)$ Torus $T^2$
Square $(0,y) \sim (1, 1-y)$ Möbius strip
Square Both pairs with twist Klein bottle
Disk $D^2$ $\partial D^2$ collapsed to a point $S^2$
$S^n$ antipodal $x \sim -x$ $\mathbb{RP}^n$

[!tip] Connection to your work Quotient topologies are the natural setting when an incontractibility partition on histories induces a coarser space — the strategy space modulo “indistinguishable to a contract.” Continuity claims about mechanisms over those classes are quotient-topology claims. Worth bookmarking next to your Cooperation Gap partition machinery.

The Separation Axioms ($T_0 \to T_4$)

These measure how well opens distinguish points. The hierarchy is strictly increasing in restrictiveness.

Hierarchy

Axiom Name Condition
$T_0$ Kolmogorov $\forall x \ne y$, $\exists$ open separating one from the other
$T_1$ Fréchet Singletons $\{x\}$ are closed
$T_2$ Hausdorff $\forall x \ne y$, $\exists$ disjoint opens $U \ni x, V \ni y$
$T_3$ Regular $T_1$ + can separate point from closed set by disjoint opens
$T_4$ Normal $T_1$ + can separate any two disjoint closed sets by disjoint opens

Hausdorff is the line in the sand

$T_2$ is where things start behaving like you expect:

  • Limits are unique.
  • Compact sets are closed.
  • The diagonal $\Delta \subseteq X \times X$ is closed.

Below $T_2$, e.g. the Zariski topology on $\text{Spec}(R)$, none of these need hold — and that’s why algebraic geometers had to invent schemes.

[!note] Counterexample to lean on The Sierpiński space $\{0, 1\}$ with opens $\{\emptyset, \{1\}, \{0,1\}\}$ is $T_0$ but not $T_1$. It is the right object for “semi-decidable predicates” in domain theory — and it appears whenever computability meets topology.

Urysohn’s Lemma

If $X$ is normal ($T_4$) and $A, B \subseteq X$ are disjoint closed sets, then there exists a continuous $f : X \to [0, 1]$ with $f|_A = 0$ and $f|_B = 1$.

This is the bridge from set-theoretic separation to function-theoretic separation. Without it, you cannot do partitions of unity, and analysis on manifolds collapses.

Tietze Extension

If $X$ is normal and $A \subseteq X$ is closed, every continuous $f: A \to \mathbb{R}$ extends to a continuous $\tilde f: X \to \mathbb{R}$.

Tightly equivalent to Urysohn. Crucial in extending sections, partial functions, etc.

Countability Axioms

These are about how “small” a description of opens you need locally and globally.

First / Second Countable, Separable

First countable: every point has a countable neighborhood basis. (Metric spaces are first countable.) Second countable: the whole topology has a countable basis. (Strictly stronger.) Separable: there exists a countable dense subset.

For metric spaces, second countable ⟺ separable ⟺ Lindelöf (every open cover has countable subcover). These split apart for general topologies.

[!note] Why first countability matters In first-countable spaces, sequences suffice to detect closure and continuity. In general topological spaces, they do not — you need nets or filters (see below). This is a recurring trap.

Compactness

The single most important property in topology.

Definition (Heine-Borel style)

$X$ is compact if every open cover $\{U_\alpha\}$ admits a finite subcover. Closed-and-bounded is the metric special case; the open-cover definition is the right one in general.

Heine-Borel Theorem

In $\mathbb{R}^n$ (with Euclidean topology), a set is compact iff it is closed and bounded.

This fails in infinite-dimensional Banach spaces — the closed unit ball is not compact in $\ell^2$ (Riesz’s lemma). This is why functional analysis needs weak topologies.

Continuous image of compact is compact

If $f: X \to Y$ continuous, $X$ compact, then $f(X) \subseteq Y$ compact.

Immediate corollary: a continuous real function on a compact space attains its maximum (extreme value theorem).

Compactness + Hausdorff = Normal

Every compact Hausdorff space is normal ($T_4$).

This is why “compact Hausdorff” is the gold-standard category — you automatically get Urysohn, Tietze, partitions of unity. Differentiable manifolds, when compact, sit here.

Bonus property: in compact Hausdorff, a bijective continuous map is automatically a homeomorphism (its inverse is continuous “for free”). This is a routine technique for proving things are homeomorphisms.

Tychonoff’s Theorem

An arbitrary product $\prod_{i \in I} X_i$ of compact spaces is compact (in the product topology).

Equivalent to the Axiom of Choice (and uses Zorn or ultrafilters in any proof). Underpins, among many things, the Banach-Alaoglu theorem in functional analysis and compactness arguments in model theory and game theory (compactness of strategy spaces in infinite games).

Compactness vs Sequential Compactness

Sequentially compact: every sequence has a convergent subsequence.

In metric spaces: compact ⟺ sequentially compact ⟺ complete + totally bounded. In general: they are incomparable. Both compact-not-sequentially-compact and sequentially-compact-not-compact exist (e.g. $[0,1]^{[0,1]}$ and the long line, respectively).

Convergence Beyond Sequences

Nets and Filters

In a non-first-countable space, sequences fail to characterize closure: there may be $x \in \bar A$ with no sequence in $A$ converging to $x$.

A net is a function $\phi : D \to X$ where $D$ is a directed set (preorder with upper bounds for pairs). Generalizes sequences (directed set $\mathbb{N}$) and lets you index by, e.g., open neighborhoods ordered by reverse inclusion.

A filter $\mathcal{F}$ on $X$ is a non-empty family of subsets closed under finite intersection and upward closure, with $\emptyset \notin \mathcal{F}$. An ultrafilter is a maximal filter. Filters/ultrafilters generalize “tail behavior” and are dual to nets.

[!note] Pragmatic takeaway Theorems “sequences detect $X$” are first-countable theorems. In general, replace sequences with nets/filters and the proofs go through (compactness ⟺ every net has a convergent subnet ⟺ every ultrafilter converges).

Connectedness, Extended

You already have the disjoint-opens definition and $[0,1]$ connected.

Components

The connected component of $x$ is the largest connected set containing $x$ — equivalently, the union of all connected sets containing $x$. Components partition $X$ and are always closed (not always open: e.g. $\mathbb{Q}$ has singleton components, none open).

Path components partition $X$ via the equivalence $x \sim y$ iff there is a path. Path components $\subseteq$ components. They coincide for locally path-connected spaces.

Totally Disconnected

$X$ is totally disconnected if components are singletons. Canonical examples: $\mathbb{Q}$, $p$-adic numbers $\mathbb{Z}_p$, the Cantor set $\mathcal{C}$.

[!note] Cantor set as universal object Every compact metric totally disconnected space without isolated points is homeomorphic to $\mathcal{C}$. It is also homeomorphic to $\{0,1\}^\mathbb{N}$ — the natural state space for infinite binary processes. This is one reason it appears in the formalization of stochastic strategies and infinite trees in game theory.

Locally (Path) Connected

Each point has a neighborhood basis of (path-)connected opens. This is what you need for path components to equal components, and for covering space theory (next) to behave.

Metrization

When can a topology be re-derived from a metric? This is the inverse of the metric-induced-topology direction.

Urysohn Metrization

A second-countable Hausdorff topological space is metrizable iff it is regular ($T_3$).

So for second-countable spaces, separation up to regularity is exactly the metrization condition.

Nagata–Smirnov

$X$ is metrizable iff $X$ is regular and has a $\sigma$-locally finite basis.

The general necessary-and-sufficient condition without assuming second countability. The takeaway: metrizability is a separation + combinatorial-size condition.

Algebraic Topology in Five Steps

The shift: classify spaces up to homeomorphism (or weaker, homotopy) by algebraic invariants.

Homotopy

Two continuous maps $f, g : X \to Y$ are homotopic, $f \simeq g$, if there is a continuous $H : X \times [0,1] \to Y$ with $H(\cdot, 0) = f$, $H(\cdot, 1) = g$. Two spaces are homotopy equivalent if there exist $f: X \to Y, g: Y \to X$ with $gf \simeq \mathrm{id}_X$, $fg \simeq \mathrm{id}_Y$.

Homotopy equivalence is strictly weaker than homeomorphism. $\mathbb{R}^n$ is homotopy equivalent to a point (contractible), though not homeomorphic to one.

Fundamental Group $\pi_1(X, x_0)$

The set of homotopy classes of loops $\gamma : [0,1] \to X$ with $\gamma(0) = \gamma(1) = x_0$, under concatenation. This forms a group.

$$ \pi_1(X, x_0) = \{ [\gamma] : \gamma \text{ loop at } x_0 \} / \simeq $$

Functorial: a continuous $f : (X, x_0) \to (Y, y_0)$ induces a homomorphism $f_* : \pi_1(X, x_0) \to \pi_1(Y, y_0)$.

Key computations:

  • $\pi_1(S^1) = \mathbb{Z}$ (the “winding number” theorem).
  • $\pi_1(S^n) = 0$ for $n \ge 2$ (simply connected).
  • $\pi_1(T^2) = \mathbb{Z}^2$.
  • $\pi_1(\mathbb{RP}^n) = \mathbb{Z}/2\mathbb{Z}$ for $n \ge 2$.

The winding-number computation immediately yields a topological proof of the Fundamental Theorem of Algebra and of Brouwer Fixed Point Theorem in dimension 2.

Covering Spaces

A covering map $p: \tilde X \to X$ is one where every $x \in X$ has an open neighborhood $U$ with $p^{-1}(U)$ a disjoint union of opens each homeomorphic to $U$ via $p$. Examples: $\mathbb{R} \to S^1$ via $t \mapsto e^{2\pi i t}$; $S^n \to \mathbb{RP}^n$ via antipodal quotient.

The Galois correspondence for covering spaces: subgroups of $\pi_1(X)$ ↔ connected covering spaces of $X$.

The universal cover $\tilde X$ is simply connected and covers every other cover. Beautiful parallel to Galois theory in algebra.

Homology (sketch)

$\pi_1$ is non-abelian and hard to compute past dimension 1. Singular homology $H_n(X; \mathbb{Z})$ assigns to each $X$ an abelian group per dimension $n$, with $H_0$ counting components, $H_1 = \pi_1^{\text{ab}}$ (abelianization), and $H_n$ for $n \ge 2$ detecting “$n$-dimensional holes.”

For spheres: $H_n(S^k) = \mathbb{Z}$ if $n \in \{0, k\}$, else $0$. This single computation gives:

  • $\mathbb{R}^n \not\simeq \mathbb{R}^m$ for $n \ne m$ (invariance of domain).
  • The Brouwer fixed point theorem in every dimension.

Euler Characteristic

For finite CW complexes: $\chi(X) = \sum_n (-1)^n \dim H_n(X; \mathbb{Q})$. For a polyhedron this gives back your Euler formula $v - e + f = \chi$. For surfaces:

Surface $\chi$ $\pi_1$
$S^2$ $2$ $0$
Torus $T^2$ $0$ $\mathbb{Z}^2$
Genus-$g$ surface $2 - 2g$ $\langle a_1, b_1, \dots, a_g, b_g \mid \prod [a_i, b_i] \rangle$
$\mathbb{RP}^2$ $1$ $\mathbb{Z}/2$
Klein bottle $0$ non-abelian

This connects directly to your existing Euler’s Theorem note — Legendre’s spherical-excess proof is the $g = 0$ case of a vast machine.

Manifolds

Topological Manifold

$M$ is an $n$-manifold if it is second-countable, Hausdorff, and locally homeomorphic to $\mathbb{R}^n$. Adding a compatible smooth atlas gives a smooth manifold; adding a Riemannian metric gives geometry.

The second-countable + Hausdorff conditions exclude pathologies like the line with two origins and the long line. These are non-paracompact and break partitions of unity.

Classification of Surfaces (revisited)

Compact connected surfaces without boundary, up to homeomorphism:

Class Orientable $\chi$
Sphere $S^2$ Yes $2$
Genus-$g$ surface $\Sigma_g$ Yes $2 - 2g$
$N_k = \mathbb{RP}^2 \# \cdots \# \mathbb{RP}^2$ ($k$ copies) No $2 - k$

The $(g, k)$ classification is complete and computable via $\chi$ + orientability. No higher-dimensional analog: classification in dimension $\ge 4$ is undecidable (Markov, 1958), connected to the word problem in group theory.

[!note] Decidability boundary Dimensions $\le 3$: classifiable. Dimension $\ge 4$: undecidable. There is something deep here, of a flavor adjacent to your computability of moral reasoning threads — formalization is bounded by group-theoretic word problems, an ambient computability ceiling.

Fixed-Point Theorems — Bridge to Game Theory

This is where the topology you’ve collected becomes directly a game-theoretic tool.

Brouwer Fixed Point Theorem

Any continuous map $f : D^n \to D^n$ from the closed $n$-disk to itself has a fixed point: $\exists x : f(x) = x$.

Equivalent formulations: the sphere $S^{n-1}$ is not a retract of $D^n$; $H_{n-1}(S^{n-1}) \ne 0$. Sperner’s lemma gives a combinatorial proof.

Kakutani Fixed Point Theorem

Let $X \subseteq \mathbb{R}^n$ be non-empty, compact, convex. Let $\Phi : X \rightrightarrows X$ be a correspondence (set-valued) with $\Phi(x)$ non-empty, convex, and with closed graph. Then $\exists x : x \in \Phi(x)$.

Why these matter for your work

Nash’s existence theorem is Brouwer applied to the best-response map on the simplex of mixed strategies — or, in the version for correspondences, Kakutani applied to the best-response correspondence (since the argmax is generally set-valued).

The topology of the mixed-strategy space (compactness, convexity, continuity of expected utilities in the mixture weights) is what makes equilibrium exist. Any time you reach for an existence result for an equilibrium concept in a richer setting.

[!tip] A research-grade hook Your No Mechanism Sufficiency Corollary lives in this neighborhood: when the contractible quotient becomes too coarse, the relevant best-response correspondence loses upper hemicontinuity, and Kakutani’s hypotheses break — which is a topological way to read mechanism incompleteness.

Banach Fixed Point Theorem

For metric (not just topological) spaces: a contraction on a complete metric space has a unique fixed point, and iteration converges geometrically. This is the constructive cousin — Brouwer is existence-only, Banach gives an algorithm. The contrast (existence-only vs. constructive convergence) is itself a recurring theme in mechanism design.

Borsuk–Ulam

Any continuous $f : S^n \to \mathbb{R}^n$ identifies some antipodal pair: $\exists x : f(x) = f(-x)$.

Corollaries: ham-sandwich theorem, Lusternik–Schnirelmann results, and — surprisingly — fair division and necklace splitting in combinatorics. The bridge from topology to combinatorial fairness is the Borsuk–Ulam family.

Two Theorems That Punch Above Their Weight

Baire Category Theorem

A complete metric space (or a locally compact Hausdorff space) is not a countable union of nowhere-dense sets.

Equivalently: a countable intersection of dense open sets is dense. The hidden engine behind the open mapping theorem, closed graph theorem, and uniform boundedness in functional analysis. Whenever someone in analysis says “generic,” they probably mean “comeager” (complement of meager), which is a Baire-category statement.

Invariance of Domain

If $U \subseteq \mathbb{R}^n$ is open and $f : U \to \mathbb{R}^n$ is continuous and injective, then $f(U)$ is open and $f$ is a homeomorphism onto its image.

Sounds obvious; isn’t. Implies $\mathbb{R}^n \cong \mathbb{R}^m \Rightarrow n = m$, and is foundational for the well-definedness of manifold dimension.

A Curiosity to Close

Banach–Tarski

Using the Axiom of Choice, a solid unit ball in $\mathbb{R}^3$ can be partitioned into finitely many pieces, which can be reassembled by rigid motions into two solid unit balls.

Not a paradox of geometry — a statement about the failure of finite additivity for non-measurable sets under the action of a free subgroup of $SO(3)$. Sets the conceptual boundary of what “measurable” topology will allow you to do.

Suggested Mental Map

Layer Concept Slogan
Generation Basis, subbasis, product, quotient “Topologies are universal constructions”
Separation $T_0 \to T_4$, Urysohn, Tietze “How much can opens distinguish?”
Size First/second countable, separable “How small a description?”
Compactness Tychonoff, Heine-Borel “Continuous + compact = good”
Convergence Nets, filters “Sequences are not enough”
Invariants $\pi_1$, $H_*$, $\chi$ “Classify up to homotopy”
Fixed points Brouwer, Kakutani, Banach “Existence theorems for equilibria”

The single takeaway: topology is the minimal structure for talking about continuity and convergence, and most of the deep theorems are about how separation + compactness + countability conspire to make a space behave. Everything else (algebraic topology, manifold theory, fixed-point theorems) is a payoff from that setup.