Royal Road to Topology Chapter I

Chapter I. Preliminaries

This chapter outlines the basic set-theoretic concepts, relations, maps, orders, polarities, and cardinalities assumed throughout.

I.1. Relations and Maps

Basic Notations

Relations

[!NOTE] Proposition I.1.1 For $R \subset X \times Y$, $A \subset X$, and $B \subset Y$, the following statements are equivalent:

$RA \cap B \neq \emptyset$

$A \cap R^- B \neq \emptyset$

$R \cap (A \times B) \neq \emptyset$

Inverse Relation
Types of Relations (for $R \subset X \times X$):
Identity Relation (Diagonal)

Maps (Functions)

[!NOTE] Proposition I.1.2 A relation $F \subset X \times Y$ is a graph if and only if $F^-Y = X$ and $F^-y_0 \cap F^-y_1 \neq \emptyset \implies y_0 = y_1$.

Injectivity, Surjectivity, Bijectivity
Permutation
Inverse Map
Restriction
Semi-Inverse Map (Definition I.1.4): If $f : X \to Y$ is injective, then $f^{\sim 1} : f(X) \to X$ defined by $f^{\sim 1}(f(x)) := x$ for each $x \in X$ is the semi-inverse map of $f$.
Extensions to Relations (Definition I.1.5): A relation $R \subset V \times W$ is:
- Surjective if $RV = W$.
- Injective if $Rv_0 \cap Rv_1 \neq \emptyset \implies v_0 = v_1$.
Fiber

[!NOTE] Corollary I.1.6 A relation $F \subset X \times Y$ is a graph if and only if its inverse relation $F^-$ is surjective and injective.

I.2. Equivalence Relations

[!NOTE] Proposition I.2.1 There exists a bijection between equivalence relations on a non-empty set $X$ and surjective maps defined on $X$.

I.3. Order

[!NOTE] Proposition I.3.8 An order isomorphism $f$ preserves extrema: $$f\left(\bigvee A\right) = \bigvee f(A), \quad f\left(\bigwedge A\right) = \bigwedge f(A)$$

Minimal and Maximal Elements

I.4. Polarities

[!NOTE] Proposition I.4.1 The pair $(R^\circ, R^{-\circ})$ is an inverse order isomorphism (with respect to $\subset$) between $R$-bipolars and $R^-$-bipolars.

[!NOTE] Proposition I.4.2 For $R \subset X \times Y$, $A \subset X$, and $B \subset Y$, the following are equivalent:

$R^\circ A \supset B$

$A \subset R^{-\circ} B$

$A \times B \subset R$

I.5. Finite and Infinite Cardinals

[!NOTE] Proposition I.5.3 A set $X$ is countable if and only if it is either finite and non-empty, or $\text{card } X = \aleph_0$.

[!NOTE] Proposition I.5.4 & Corollary I.5.6

The countable union of countable sets is countable.

The product of two countable sets is countable.

Consequently, the set of rational numbers $\mathbb{Q}$ is countable.

Uncountability

[!NOTE] Theorem I.5.8 (Cantor) For every set $X$: $$\text{card } X < \text{card } 2^X$$