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Munkres, Section 30 The Countability Axioms

First countability axiom: for every point $x\in X$ there is a countable basis at $x$ . $X$ is called first-countable.
Continuous functions and converging sequences in first-countable spaces (compare to §21):
- Converging sequences of points and the closure (The Sequence Lemma): let $X$ be a topological space, then
  - in any topological space: if there is a sequence of points of $A\subseteq X$ converging to $x$ then $x\in\={A}$ .
  - if $X$ is first-countable: if $x\in\={A}$ then there is a sequence of points of $A$ converging to $x$ .
- Continuity and convergent sequences of points: let $f:X\rightarrow Y$ , then
  - in any topological space: if $f$ is continuous at $x$ then for every $x_n\rightarrow x$ : $f(x_n)\rightarrow f(x)$ .
  - if $X$ is first-countable (and $Y$ is an arbitrary space): if for every $x_n\rightarrow x$ : $f(x_n)\rightarrow f(x)$ then $f$ is continuous at $x$ .
Lindelöf space: every open covering has a countable subcovering.
Separable space: there is a countable dense. A dense is a subset of a space such that its closure is the whole space (every point not in it is its limit point).
Second countability axiom: $X$ has a countable basis for its topology. $X$ is said to be second-countable.
- A second countable space is both Lindelöf and separable.
- If a metric space is Lindelöf or separable then it is second countable.
Subspaces and countability axioms:
- A subspace of a first-countable (second-countable) space is first-countable (second-countable).
- A subspace of a separable (Lindelöf) space need NOT be a separable (Lindelöf) space.
  - An open subspace of a separable space is separable.
  - A closed subspace of a Lindelöf space is Lindelöf.
- If a subspace of a second-countable space is discrete then it is countable.
- If a subset of a second-countable space is uncountable then only countably many points in the subset are not its limit points.
Products and countability axioms:
- A countable product of separable (first-countable, second-countable) spaces (in the product topology) is separable (first-countable, second-countable).
- A product of two Lindelöf spaces does NOT have to be Lindelöf.
  - Lindelöf $\times$ compact is Lindelöf.
- A continuum product of separable spaces is separable.
- If the product of any number of spaces possesses one of these 4 properties, so does each space in the product.
Continuous functions and countability axioms:
- The continuous image of a first-countable space need NOT be first-countable.
- The continuous (even the quotient) image of a second-countable space need NOT be second-countable.
  - The continuous open image of a first-countable (second-countable) space is first-countable (second-countable).
- The continuous image of a separable (Lindelöf) space is separable (Lindelöf).
Topological groups and countability axioms:
- If a topological group is first-countable and (separable or Lindelöf) then it is second-countable.

		separable			not separable
		Lindelöf		not Lindelöf	Lindelöf		not Lindelöf
		compact	not compact	not compact	compact	not compact	not compact
first- countable	second- countable	$[0,1]^2$	$\mathbb{R}$
first- countable	not second- countable	double arrow: $D=$ $([0,1]\times\{0,1\})_{dic}$ $D^2$	$\mathbb{R}_l$	$\mathbb{R}_l^2$	$[0,1]^2_{dic}$		$\mathbb{R}_l^2\cap\{y\ge -x\}$ $S_\Omega$
not first- countable	not second- countable	$[0,1]^{[0,1]}$	$\mathbb{Q}^\infty_{box}$ (eventually 0)		$\={S}_\Omega$

Munkres, Section 31 The Separation Axioms

Regular space: a $T_1$ -space such that a closed subset and a point not in it can be separated by two open sets.
- A space is regular iff it is $T_1$ and any neighborhood of a point contains the closure of a neighborhood of the point.
Normal space: a $T_1$ -space such that any two closed disjoint subsets can be separated by two open neighborhoods.
- A space is normal iff it is $T_1$ and any neighborhood of a closed set contains the closure of a neighborhood of the set.
Refining of topology and separation axioms:
- Neither regularity nor normality of the space is refining-preserved property.
Subspaces and separation axioms:
- A subspace of a regular space is regular.
- A subspace of a normal space need NOT be normal.
  - A closed subspace of a normal space is normal.
Products and separation axioms:
- The product of regular spaces is regular.
- The product of even two normal spaces need NOT be normal.
- If the product of any number of spaces is Hausdorff (regular, normal), so is each space in the product.
Continuous functions and separation axioms:
- A closed continuous image of a normal space is normal.
- The image of a *-space under a perfect map (see §26) is a *-space, where * stands for either one:
  - Hausdorff
  - regular
  - locally compact
  - second-countable
Topological groups and separation axioms: let $G$ be a topological group, $X$ be a space, then
- An action $\alpha$ of $G$ on $X$ is a map $G\cdot X\to X$ such that 0) it is continuous, i) $e\cdot x=x$ (do nothing), ii) g\cdot(g’\cdot x)=(g\cdot g’)\cdot x.
- The orbit space of the action $\alpha$ is the quotient space denoted by $X|G$ given by $x~g\cdot� x$ , $g\in G$ .
- If $G$ is compact then the quotient map is a perfect map.
  - If $G$ is compact and $X$ is Hausdorff (regular, normal, locally compact, second-countable), so is $X|G$ .

Munkres, Section 32 Normal Spaces

Hausdorff + Compact $\Rightarrow$ Normal
- Hausdorff + Locally Compact $\Rightarrow$ Regular (Completely Regular: exercise 7 of §33).
- Compact implies a closed subset is compact; in a Hausdorff space two compacts can be separated.
Regular + Lindelöf $\Rightarrow$ Normal
- Cover each of two closed sets with a countable collection of open sets (Lindelöf) such that their closures do not intersect the other set (regular); subtract from nth open set the union of closures of all open sets from 1 to n covering the other set.
Metric $\Rightarrow$ Normal
- Cover each point in two disjoint closed sets with a ball such that if your double its radius it still does not intersect the other set. That’s the beauty of a metric!
Ordered $\Rightarrow$ Normal (in the order topology)
- The product of two ordered (even well-ordered) spaces need NOT be normal: $S_\Omega\times\={S}_\Omega$ is not normal.
- Well-ordered: (a,b]=(a,b+1) are open and form a basis, cover each closed set with such intervals that do not intersect the other set.
- General case (ordered): covered, for example, in Steen, Seebach, Counterexample 39, 1-6.
A completely normal space is a space such that every its subspace is normal.
- A pair of subsets is separated (wtf? what is not called separated or separation?) if they are disjoint and neither one contains a limit point of the other.
- A space is completely normal iff every pair of separated subsets can be separated by neighborhoods.
Subspaces and complete normality:
- A subspace of a completely normal space is completely normal.
Products and complete normality:
- The product of even two completely normal spaces needs NOT to be normal.
Completely normal spaces:
- a regular second-countable space
- a metric space
- an ordered space

Munkres, Section 33 The Urysohn Lemma

Two subsets $A,B\subseteq X$ are said to be separated by a continuous function if there is a continuous function $f:X\to[0,1]$ such that $f(A)=\{0\}$ and $f(B)=\{1\}$ .
Urysohn Lemma: if $X$ is normal and $A,B\subseteq X$ are closed and disjoint, then they can be separated by a continuous function.
- Consider $U_p=\{x:f(x)<p\}$ and $U_p'=\{x:f(x)\le p\}$ : the first is open and contained in the second which is closed, moreover, for every $q>p$ : $U_p'\subseteq U_q$ . We use the normality of the space to construct a sequence of such open sets and define $f(x)=\inf\{p:x\in U_p\}$ .

Why cannot the proof of the Urysohn lemma be generalized to show that in a regular space… you can also separate points from closed sets by continuous functions? At first glance, it seems that the proof of the Urysohn lemma should go through… Requiring that one be able to separate a point from a closed set by a continuous function is, in fact, a stronger condition…

Completely regular space: each one point set is closed and can be separated from a disjoint closed set by a continuous function.

In the early years of topology, the separation axioms… were labelled $T_1$ , $T_2$ (Hausdorff), $T_3$ (regular), $T_4$ (normal), and $T_5$ (completely normal)… The letter “T” stands for the German “Trennungsaxiom,” which means “separation axiom.” Later, when the notion of complete regularity was introduced, someone suggested facetiously that it should be called the “ $T_{3\frac{1}{2}}$ axiom”… The terminology is in fact sometimes used in the literature!

Subspaces and complete regularity:
- A subspace of a completely regular space is completely regular.
- If $A,B$ are disjoint closed sets and one of them is compact, they can be separated by a continuous function.
Products and complete regularity:
- The product of any family of completely regular spaces is completely regular.
A connected regular space with at least two points is uncountable.
- But there is a countably infinite connected Hausdorff space.
Perfectly normal space: a normal space such that every closed set is a $G_\delta$ -set: the intersection of a countably many open sets.
- $f:X\Rightarrow\mathbb{R}_+$ is said to vanish precisely on $A$ if $A=f^{-1}(\{0\})$ .
- The Strong Urysohn Lemma: every two disjoint closed $G_\delta$ subsets $A$ and $B$ of a normal space $X$ can be separated by a continuous function $f:X\to[0,1]$ such that $f$ vanishes precisely on $A$ and $1-f$ vanishes precisely on $B$ .
- $X$ is perfectly normal if and only if it is $T_1$ and for every closed set there is a continuous function that vanishes precisely on the set.
- Perfectly normal implies completely normal.
Completely regular and perfectly normal spaces:
- A topological group is completely regular.
- A metric space is perfectly normal.

Munkres, Section 34 The Urysohn Metrization Theorem

A family of continuous functions $\{f_\alpha\}$ separates points from closed sets if for every closed set and a point not in it there is a function in the family such that it is constant on the closed set and takes a different value at the point.
The Imbedding Theorem: If $X$ is $T_1$ and $\{f_\alpha\}_{\alpha\in J}$ separates points and closed sets in $X$ then $(f_\alpha(x))_{\alpha\in J}$ is an embedding of $X$ in $\mathbb{R}^J$ . If all functions map $X$ to $[0,1]$ then the embedding is in $[0,1]^J$ .
A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^J$ for some $J$ .
- $\Rightarrow$ The Imbedding Theorem. $\Leftarrow$ A subspace of the product of completely regular spaces is completely regular.
The Urysohn Metrization Theorem: Regular + Second-countable $\Rightarrow$ Metrizable.
- regular + second-countable implies normal + second-countable implies for a countable collection of pairs of basis neighborhoods such that $\={B}_m\subseteq B_n$ we can find a continuous function $g_{n,m}$ equal 1 inside $\={B}_m$ and 0 outside $B_n$ : this is a countable family of continuous functions that separates points from closed sets.
- A regular second-countable space is homeomorphic to a subspace of the infinite-dimensional euclidean space $\mathbb{R}^\omega$ .
Some facts about metrization that follow:
- A second-countable space is metrizable iff it is regular.
- A compact Hausdorff space is metrizable iff it is second-countable.
  - A compact Hausdorff space that is the union of two closed metrizable subspaces is metrizable.
- The one-point compactification of a locally compact Hausdorff space $X$ is metrizable iff $X$ is second-countable.
  - So, if a locally compact Hausdorff space is second-countable then it is metrizable, but not vice versa: discrete uncountable.
A locally metrizable space is a space such that every point has a metrizable neighborhood (in the subspace topology).
- A compact Hausdorff space is metrizable iff it is locally metrizable.
- A Lindelöf regular space is metrizable iff it is locally metrizable.
  - $\mathbb{R}_K$ is Lindelöf Hausdorff and locally metrizable, yet not metrizable.

Hausdorff	regular	completely regular	normal	completely normal	perfectly normal
$\mathbb{R}_K$	p.214	$\mathbb{R}_l^2$ $(0,1)^{[0,1]}$ $S_\Omega\times\={S}_\Omega$ $\mathbb{R}^J$ , $J>\omega$ $\mathbb{R}^J_{box}$ , $J>\omega$ topological_group locally_compact_Hausdorff	$[0,1]^{[0,1]}$ $\={S}_\Omega^2$ $\mathbb{R}^\omega_{box}$ (?: if the continuum hypothesis is assumed) Lindelöf_regular compact_Hausdorff	$\={S}_\Omega$ ordered_space	$\mathbb{R}_l$ $\mathbb{R}^\omega$ $\mathbb{R}^\omega_{uniform}$ metric_space second_countable_regular

Munkres, Section 35* The Tietze Extension Theorem

Tietze Extension Theorem: $X$ is normal, $A$ is closed in $X$ , $f:A\to B$ where $B=[0,1]$ or $B=\mathbb{R}$ is continuous, then $f$ can be extended to a continuous function $X\to B$ .
- The idea: if the range of a function is [-r,r] using the Urysohn lemma construct a continuous function such that its range is [-r/3,r/3] and it is never more than 2r/3 from the original function, then take the difference and do it again.
A retract is a subspace of a topological space such that there exists a surjective retraction onto it (see the Supplementary exercises of Chapter 2).
- A retraction is a quotient map.
- A retract of a Hausdorff space is a closed subspace.
A normal space is an absolute retract if whenever a closed subspace of a normal space is homeomorphic to it, the subspace is a retract.
A space $Y$ has the universal extension property if every function that maps continuously a closed subset of a normal space into $Y$ can be extended onto the whole space.
- If $Y$ is normal then it has the universal extension property iff it is an absolute retract.
- A space homeomorphic to a retract of $\mathbb{R}^J$ has the universal extension property.
The adjunction space $Z$ for spaces $X$ , $Y$ and a continuous function $f:A\to Y$ where $A\subseteq X$ , is the quotient space of the space $X\cup Y$ obtained by identifying a point $y$ with all points in $f^{-1}(y)$ . Let $p$ be the quotient map.
- If $A$ is closed, $Z$ is Hausdorff.
- $Y$ is homeomorphic to $p(Y)$ , and if $A$ is closed then $p|Y$ is a closed imbedding.
- $X-A$ is homeomorphic to $p(X-A)$ , and if $A$ is closed then $p|(X-A)$ is an open imbedding.
- If $A$ is closed, $X$ and $Y$ are normal, then the adjunction space is normal.
The coherent topology: given a sequence of spaces $X_i\subseteq X_{i+1}$ such that $X_i$ is closed in $X_{i+1}$ we define the topology on their union $X$ coherent with the subspaces $X_n$ : $U$ is open in $X$ iff $U\cap X_n$ is open in $X_n$ for every $n$ .
- $X_i$ is a closed subspace of $X$ .
- If $f|X_i$ is continuous for every $i$ then $f$ is continuous.
- If every $X_i$ is normal then $X$ is normal.

Munkres, Section 36* Imbeddings of Manifolds

An m-manifold is a Hausdorff second-countable space such that every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^m$ .
- being Hausdorff is not a local property, and without requiring it an m-manifold does need to be Hausdorff; however, being a $T_1$ -space is a local property, and, in fact, if we did not require a manifold to be a Hausdorff, it would be at least $T_1$
- second-countability requirement is needed for the theorem that states that it can be imbedded into an euclidean space (obviously, both the Hausdorff property and second-countability are necessary, but they turn out to be sufficient): see below
- if the space is compact and satisfies all the properties in the definition of an $m$ -manifold except for second-countability then it must be second-countable and, hence, $m$ -manifold: see below
- a manifold is regular (given that it is Hausdorff), therefore, metrizable (being regular and second-countable)
The support of $\phi:X\to\mathbb{R}$ is the closure of $\phi^{-1}(\mathbb{R}-\{0\})$ .
A point-finite collection of subsets is a collection of subsets such that every point is contained in a finite number of sets in the collection.
- Compare to a stronger notion of a locally finite collection of subsets (Section 17).
A partition of unity: a set of functions $\{\rho_j:X\to[0,1]\}$ such that for every $x\in X$ only a finite number of $\rho_j(x)\neq 0$ and the sum of all $\rho_j(x)$ equals $1$ .
- A partition of unity is said to be dominated by an open covering $\{U_j\}$ indexed over the same index set if for every $j$ , the support of $\rho_j$ is contained in $U_j$ .
- If $X$ is normal then for every finite collection of open sets covering $X$ there is a partition of unity dominated by the open covering.
  - In fact, the result holds true for every countable point-finite collection of open sets covering the normal space.
If space is a $m$ -manifold then it can be imbedded in a $n$ -dimensional euclidean space where $n\in\mathbb{Z}_+$ .
- This chapter proves the result for compact spaces only.
- We can cover the space by a finite collection of n open sets such that each is homeomorphic to the m-dimensional euclidean space, then find a partition of unity dominated by the collection; then we map each point to a point in the (n+nm)-dimensional euclidean space by taking the values at the point of the functions in the partition of unity and also by taking their products with the homeomorphisms (or zero if the homeomorphism is not defined for the point — this is why we need to take the product, so that the resulting function is continuous); this mapping is continuous and injective (the partition of unity part of the resulting vector ensures that points not in the same open set have different coordinates, while the other part results in different coordinates for points in the support of the same function in the partition of unity).
Compact Hausdorff spaces and manifolds: let $X$ be compact and Hausdorff, then
- If each point $x$ has a neighborhood that can be imbedded into $\mathbb{R}^{k_x}$ for some $k_x$ then $X$ can be imbedded into some finite-dimensional euclidean space as well.
- If for some $m$ each point $x$ has a neighborhood homeomorphic to an open subset of $\mathbb{R}^m$ then $X$ is an $m$ -manifold.