Cardinal Functions of Topological Spaces

#Topology

Subjects: Topology
Links: Topological Spaces, Arithmetic of Cardinal Numbers, Regular and Singular Cardinals, Bases, Subbases, and Local Basis for Topological Spaces, Dense Subsets

Weight and Character

Def: Let $(X, τ)$ be a topological space. We define the weight of space $(X, τ)$ is the $$w((X, \tau)) := \min{|\mathcal B| \mid \mathcal B \subseteq \tau, \text{ where }\mathcal B \text{ is base for }\tau}.$$If the topology is clear, then it is denoted $w (X)$ .

Def: The character of a point $x$ in a topological space $(X, τ)$ is defined as $$\chi(x, (X, \tau)) := \min{|\mathcal B(x)| \mid B(x) \subseteq \tau, \text{ where }\mathcal B \text{ is a local base for }X \text{ at } x}. $$If the topology is clear is denoted as $χ (x, X)$ . The character of a topological space $(X, τ)$ is defined as $$\chi(X, \tau) := \sup{\chi(x, (X, \tau)) \mid x \in X}.$$If the topology is clear, then it is denoted as $χ (X)$ . Additionally, we can define the character for a set $A \subseteq X$ as $$\chi(A, (X, \tau)) := \min{|\mathcal B(x)| \mid B(x) \subseteq \tau, \text{ where }\mathcal B \text{ is a local base for }X \text{ at }A}. $$

Obs: If we have that $χ (X, τ) \leq ℵ_{0}$ then we note that this equivalent to the space being first countable. If we have that $w (X, τ) \leq ℵ_{0}$ this is equivalent to the space being second countable.

Th: If $w (X) \leq μ$ , then for every family ${U_{α} ∣ α < κ} \subseteq τ$ there exists a set $S \subseteq κ$ such that $| S | \leq μ$ and $$\bigcup_{\alpha \in S} U_\alpha = \bigcup_{\alpha < \kappa} U_\alpha.$$
Th: If $w (X) \leq μ$ then for every $B$ for $X$ there exists a $B_{0}$ such that $| B_{0} | \leq μ$ and $B_{0} \subseteq B$ .

Th: If $f : X \to Y$ is an open mapping, then for every $x \in X$ and $A \subseteq X$ we have $χ (f (x), Y) \leq χ (x, X)$ and $χ (F [A], Y) \leq χ (A, X)$ . If, moreover, $f$ is surjective, then $w (Y) \leq w (X)$ , and $χ (Y) \leq χ (X)$ .

Th: For every Kolmogorov space we have $| X | \leq 2^{w (X)}$ .

Obs: Let $M$ be a subspace of $X$ . If $A \subseteq M$ , and $x \in M$ , then $χ (A, M) \leq χ (A, X)$ , and $χ (x, M) \leq χ (x, M)$ .

Prop: If $X$ be a regular space, and $M$ is dense subset of $X$ , then any $x \in M$ will satisfy $χ (x, M) \leq χ (x, X)$ .

Cor: Let $X$ be a topological space, and $M$ be a dense in $X$ . If $A \subseteq M$ satisfies that for every closed $B \subseteq X$ that is disjoint from $A$ there are $U, V \in τ_{X}$ such that $A \subseteq U$ , $B \subseteq V$ and $U \cap V = \emptyset$ , then $χ (A, M) = χ (A, X)$ .

Cor: If $X$ is a normal space, $M$ is dense in $X$ and $A \subseteq M$ , then $χ (A, M) = χ (A, X)$ .

Prop: Let $X$ be a topological space. If $M$ is a closed subspace of $X$ , and $A \subseteq X$ , then $χ (A \cap M, M) \leq χ (A, X)$ .

Prop: If $f : X \to Y$ is a closed continuous function, and $B \subseteq Y$ , then $χ (f^{- 1} [B], X) \leq χ (B, Y)$ , additionally, if $f$ is surjective, then $χ (f^{- 1} [B], X) = χ (B, Y)$ .

Th: Every infinite $T_{2}$ compact space $X$ satisfies $| X | \leq \exp (χ (X))$

Cor: very infinite first countable $T_{2}$ compact space $X$ satisfies $| X | \leq c$ .

Psuedocharacter

Def: Let $X$ be a $T_{1}$ space. The psuedocharacter of a point $x \in X$ is defined as the smallest cardinal of the form $| U |$ , where $U \subseteq τ_{X}$ and $⋂ U = {x}$ ; this cardinal is denoted by $ψ (x, X)$ . Additionally, the psuedocharacter of $X$ is defined as $$
\psi(X) := \sup{\psi(x, X) \mid x\in X}.

**Obs:** Let $X$ be $T_1$ space. Note that for every $x\in X$, then it is satisfied that $\psi(x, X) \le \chi(x, X)$, and $\psi(X) \le \chi(X)$. **Prop:** If $X$ is a $T_0$ space, and has a $G_\delta$ diagonal, then $\psi(X) = \omega$. **Prop:** If $X$ is a $T_2$ compact space, then $\psi(x, X) = \chi(x, X)$ and $\psi(X) = \chi(X)$. # Density **Def:** The *density of a space $X$* is defined as: $$d(X):= \min\{|D| \mid D \subseteq X, D\text{ is dense in} X\}.$$If $d(X) \le \aleph_0$, then we say that $X$ is separable. **Prop:** For every topological space $X$ we have that $d(X) \le w(X)$. This actually gives us a nice proof for separable implies separable. **Th:** If there's a a continuous surjective function $f: X \to Y$, then $d(Y) \le d(X)$. **Cor:** A continuous image of separable space is separable **Th:** For everyHausdorff spacewe have that $|X| \le 2^{2^{d(X)}}$ and $|X| \le [d(X)] ^{\chi(X)}$. **Th:** For everyregular Hausdorff spacewe have $w(X) \le 2^{d(X)}$. # Network Weight **Def:** Let $(X, \tau)$ be a topological space. We define the *network weight* of $(X, \tau)$ as

nw(X) = nw(X, \tau) :=\min {|\mathcal N| \mid \mathcal N \text{ is network for }(X, \tau) }.

* * O b s : * * W e s e e t h a t e v e r y b a s e f o r $ (X, τ) $ i s a n e t w o r k a n d t h e s e t $ {{x} ∣ x \in X} $ i s a l s o a n e t w o r k, t h e n $ $ d (X, τ) \leq n w (X, τ) \leq min {w (X, τ), | X |} .

Th: For every Kolmogorov space we have $| X | \leq 2^{n w (X)}$ .

Prop: If $X$ is a $T_{2}$ space, then there are $Y$ a $T_{2}$ space and $f : X \to Y$ be a continuous bijective function such that $w (Y) \leq n w (X)$ .

Prop: If $X$ is a $T_{2}$ compact space, then $n w (X) = w (X)$ .

Cor: If $X$ is a $T_{2}$ compact space and a has a cover ${A_{α} ∣ α < κ}$ such that $n w (A_{α}) \leq λ \geq ω$ for $α < κ$ and $κ \leq λ$ , then $n w (X) \leq λ$ .

Th: For every $T_{2}$ compact space $X$ we have $w (X) \leq | X |$

Th: Let $X$ and $Y$ be $T_{2}$ spaces. If there's a continuous surjective function $f : X \to Y$ , and $Y$ is compact, then $w (Y) \leq n w (X)$ .