Urysohn Lemma and applications

The Urysohn Lemma

Theorem 421.1(Urysohn lemma).

Let $X$ be a normal space, let $A$ and $B$ be disjoint closed subsets of $X$. Let $[a,b]$ be a closed interval in the real line. Then there exists a continuous map

$$f:X\to [a,b]$$

such that $f(x)=a$ for every $x\in A$, and $f(x)=b$ for every $x\in B$.

Definition 421.2(Definition).

If $A$ and $B$ are two subsets of the topological space $X$, and if there is a continuous function $f:X\to [0,1]$ such that $f(A)=\{0\}$ and $f(B)=\{1\}$, we say that $A$ and $B$ can be separated by a continuous function.

Definition 421.3(Definition).

A space $X$ is completely regular if one-point sets are closed in $X$ and if for each point $x_0$ and each closed set $A$ not containing $x_0$, there is a continuous function $f:X\to [0,1]$ such that $f(x_0)=1$ and $f(A)=\{0\}$.

Proposition 421.4(Proposition).

A subspace of a completely regular space is completely regular. A product of completely regular spaces is completely regular.

da93d4

---

Urysohn metrization theorem

Theorem 421.5(Urysohn).

Every second countable $T_3$ topological space is metrizable (In particular, can be imbedded in ${\mathbb{R}}^{\omega }$ with the product topology).

Proposition 421.6(Proposition).

Let $X$ be a space in which one-point sets are closed. Suppose that set $\{f_{\alpha }{\}}_{\alpha \in J}$ is an indexed family of continuous functions $f_{\alpha }:X\to \mathbb{R}$ satisfying the requirement that for each point $x_0$ of $X$ and each neighborhood $U$ of $x_0$, there is an index $\alpha$ such that $f_{\alpha }$ is positive at $x_0$ and vanishes outside $U$. Then the function $F:X\to {\mathbb{R}}^J$ defined by

$$F(x)=(f_{\alpha }(x){)}_{\alpha \in J}$$

is an imbedding of $X$ in ${\mathbb{R}}^J$. If $f_{\alpha }$ maps $X$ into $[0,1]$ for each $\alpha$, then $F$ imbeds $X$ in $[0,1{]}^J$.

3206b8

Proposition 421.7(Proposition).

A space $X$ is completely regular iff it is homeomorphic to a subspace of $[0,1{]}^J$ for some $J$.

Proof.

$(⟹)$ For every pair $(x,U)$, where $x\in X$ and $U$ is an open neighborhood of $x$, let $f_{(x,U)}$ be the function which is $1$ on $x$ and zero on $\overline{U}$. Use Prp 6 with the collection $\{f_{(x,U)}\}$.

$(⟸)$ Follows immediately from Prp 4.□

---

The Tietze Extension Theorem

Theorem 421.8(Tietze).

Let $X$ be a normal space. Let $A$ be a closed subspace of $X$.

1. Any continuous map of $A$ into the closed interval $[a,b]$ of $\mathbb{R}$ may be extended to a continuous map of all of $X$ into $[a,b]$.
2. Any continuous map of $A$ into $\mathbb{R}$ may be extended to a continuous map of all of $X$ into $\mathbb{R}$.

---

Imbeddings of Manifolds

We have shown that every second countable regular space can be imbedded in the infinite dimensional euclidean space ${\mathbb{R}}^{\omega }$. Under what conditions can a space $X$ be imbedded in some finite dimensional euclidean space ${\mathbb{R}}^N$?

Definition 421.9($m$-manifold).

An $m$-manifold is a second countable Hausdorff space $X$ such that each point $x$ of $X$ has a neighborhood that is homeomorphic with an open subset of ${\mathbb{R}}^m$.

Definition 421.10(Definition).

If $\phi :X\to \mathbb{R}$, then the support of $\phi$ is defined to be the closure of the set ${\phi }^{-1}(\mathbb{R}\setminus \{0\})$. Thus, if $x$ lies outside of the support of $\phi$, there is some neighborhood of $x$ on which $\phi$ vanishes.

Definition 421.11(Definition).

Let $\{U_1,\dots ,U_n\}$ be a finite indexed open covering of the space $X$. An indexed family of continuous functions

$${\phi }_i:X\to [0,1]\text{for }i=1,\dots ,n,$$

is said to be a partition of unity dominated by $\{U_i\}$ if

1. $\text{supp}({\phi }_i)\subseteq U_i$ for each $i$.
2. ${\sum }_{i=1}^n{\phi }_i(x)=1$ for each $x$.

Proposition 421.12(Proposition).

Let $\{U_1,\dots ,U_n\}$ be a finite open covering of the normal space $X$. Then there exists a partition of unity dominated by $\{U_i\}$.

Theorem 421.13(Theorem).

If $X$ is a compact $m$-manifold, then $X$ can be imbedded in ${\mathbb{R}}^N$ for some positive integer $N$.