Topology – Complete Formula Sheet

1. Topological Spaces

Definition

Space $(X, \mathcal{T})$ where $\mathcal{T}$ satisfies:

$\emptyset, X \in \mathcal{T}$

Finite intersections in $\mathcal{T}$

Arbitrary unions in $\mathcal{T}$

Open Sets

Elements of $\mathcal{T}$ are open sets

Examples

Discrete: $\mathcal{T} = \mathcal{P}(X)$

Indiscrete: $\mathcal{T} = \{\emptyset, X\}$

Cofinite: Open if $U^c$ finite

2. Basis & Subbasis

Basis

$\mathcal{B}$ generates $\mathcal{T}$ if:

Every $x \in X$ in some $B \in \mathcal{B}$

If $x \in B_1 \cap B_2$, exists $B_3$ with $x \in B_3 \subseteq B_1 \cap B_2$

Subbasis

$\mathcal{S}$ generates $\mathcal{T}$ via finite intersections of $\mathcal{S}$

Example

Open balls $(x,r)$ form basis for metric topology

3. Closed Sets

Definition

$F$ closed iff $F^c$ open

Closure

$\bar{A} = A \cup \text{(limit points)}$

Interior

$\text{int}(A) = \bigcup\{U : U \subseteq A, U \text{ open}\}$

Boundary

$\partial A = \bar{A} \setminus \text{int}(A)$

4. Continuous Functions

Definition

$f: X \to Y$ continuous iff for open $V \subseteq Y$, $f^{-1}(V)$ open in $X$

Equivalent

$f^{-1}(F)$ closed for closed $F \subseteq Y$

Preserves convergent sequences (metrizable case)

Properties

Composition continuous

Image of compact is compact

5. Homeomorphisms

Definition

$f: X \to Y$ homeomorphism iff:

$f$ bijection

$f$ and $f^{-1}$ continuous

$X \cong Y$

Invariants

Compactness preserved

Connectedness preserved

Separation axioms preserved

6. Subspace Topology

Definition

$A \subseteq X$ with $\mathcal{T}_A = \{U \cap A : U \in \mathcal{T}\}$

Open in Subspace

$V \subseteq A$ open iff $V = U \cap A$ for $U$ open in $X$

Closed in Subspace

$F \subseteq A$ closed iff $F = C \cap A$ for $C$ closed in $X$

7. Product Topology

Definition

$X \times Y$ has basis of open rectangles $U \times V$

Box vs Product

Product: finite factors have standard form

Box: infinite factors need caution

Key Property

Compact iff each factor compact (Tychonoff)

Connected iff each factor connected

8. Quotient Topology

Definition

$U \subseteq X/\sim$ open iff $p^{-1}(U)$ open in $X$

Universal Property

$g: X/\sim \to Z$ continuous iff $g \circ p$ continuous

Examples

Circle: $\mathbb{R}/\mathbb{Z}$

Torus: $\mathbb{R}^2/\mathbb{Z}^2$

9. Metric Spaces

Metric Definition

$d: X \times X \to \mathbb{R}_{\geq 0}$ with:

$d(x,y) = 0 \iff x = y$

$d(x,y) = d(y,x)$ (symmetry)

$d(x,z) \leq d(x,y) + d(y,z)$ (triangle inequality)

Induced Topology

Open balls: $B(x,r) = \{y : d(x,y) < r\}$

Metrizable space: admits compatible metric

10. Convergence

Sequences

$(x_n) \to x$ iff every open $U \ni x$ contains all but finitely many $x_n$

Nets

Generalization: $(x_i)_{i \in I}$ directed set $I$

Useful in non-first-countable spaces

Limit Point

$x$ limit point of $A$ iff every open $U \ni x$ intersects $A \setminus \{x\}$

11. Separation Axioms

T₀ (Kolmogorov)

For distinct $x,y$: open containing exactly one

T₁

For distinct $x,y$: open $U \ni x$ with $y \notin U$, and vice versa

T₂ (Hausdorff)

For distinct $x,y$: disjoint opens $U \ni x$, $V \ni y$

T₃, T₄

T₃ (Regular): $T_0$ + regular separation

T₄ (Normal): $T_1$ + normal separation

12. Compactness

Definition

Every open cover has finite subcover

Heine-Borel

In $\mathbb{R}^n$: compact iff closed & bounded

Key Theorems

Compact + Hausdorff = T₄

Continuous image of compact is compact

Closed subset of compact is compact

13. Connectedness

Connected

Not union of two disjoint nonempty opens

Path Connected

Path between any two points

Properties

Continuous image connected

Union of connected sets with nonempty intersection

Components

Equivalence classes under "$x \sim y$ iff in same connected subset"

14. Countability Axioms

First Countable

Each point has countable basis of neighborhoods

Second Countable

Countable basis for entire topology

Separable

Countable dense subset

Order

2-countable $\Rightarrow$ separable

2-countable $\Rightarrow$ 1-countable

15. Urysohn & Tietze

Urysohn's Lemma

Normal $T_4$ space: for disjoint closed $F,G$, exists continuous $f: X \to [0,1]$ with $f(F)=\{0\}$, $f(G)=\{1\}$

Tietze Extension

Normal $T_4$: continuous $f: A \to \mathbb{R}$ extends to continuous $\tilde{f}: X \to \mathbb{R}$

Both theorems fundamental to function separation

16. Tychonoff's Theorem

Statement

Arbitrary product of compact spaces is compact

$\prod_{i \in I} X_i$ compact if each $X_i$ compact

Significance

Requires axiom of choice

Equivalent to AC in ZF set theory

Foundational in functional analysis

TOPOLOGY Formula Sheet