8. Urysohn's Metrization Theorem. Topological Properties Defined by Covers. Compact, Finally Compact, and Paracompact Spaces.

Lecture

26.6. Urysohn's Theorem. Every normal space X with a countable base is metrizable.

Proof. Since a subspace of a metrizable space is metrizable (see

Example 10.2.4 and Proposition 11.6 of Lecture 2), it suffices, by Theorem 26.5, to construct an embedding F : X → I

ℵ

0 of the space X into the Hilbert cube. Let B — be a countable base of the space X.

Call a pair of elements (V, U) of the base B normal if

Cl(V ) ⊂ U.

The set of all normal pairs, like the set of all pairs of elements of the base B, is countable. Let us number them by natural numbers:

(Vn, Un), n ∈ N.

By Urysohn's Lemma, for each normal pair (Vn, Un) there exists a continuous function

fn : X → [0, 1],

such that

fn(Cl(Vn)) = 1, fn(X \ Un) = 0.

It is easy to check that the family of functions fn, n ∈ N, satisfies the condition of Lemma 25.3 (separates points and closed sets). Then

F = ∆

n∈N

fn : X → I

N − is an embedding.

§ 27. Compact, finally compact, and paracompact spaces.

27.1. Shrinking Lemma. Let u = {U1, . . . , Uk} — be an open cover of a normal

space X. Then there exists an open cover v = {V1, . . . , Vk} of the space

X such that Cl(Vi) ⊂ Ui

, i = 1, . . . , k.

Proof. We construct the sets Vi by induction. Let F1 = X \

{Ui

: i ≥ 2}. Then

F1 ⊂ U1. By normality of the space X there exists a neighborhood OF1 such that Cl(OF1) ⊂

U1. Set V1 = OF1. From the construction it follows that the family

v1 = {V1, U2, . . . Uk} − is an open cover of X.

Suppose that we have constructed open sets V1, . . . , Vm, 1 ≤ m < k, satisfying

the conditions

Cl(Vi) ⊂ Ui

, i = 1, . . . , m;

vm = {V1, . . . , Vm, Um+1, . . . , Uk} − is an open cover of X.

Set

Fm+1 = X \ (V1 ∪ . . . ∪ Vm ∪ Um+2 ∪ . . . ∪ Uk).

Then the set Fm+1 is closed in X and lies in Um+1. There exists a neighborhood OFm+1 such that

Cl(OFm+1) ⊂ Um+1. Setting Vm+1 = OFm+1, we obtain an open cover vm+1 = {V1, . . . , Vm, Vm+1, Um+2, . . . , Uk of the space X. The cover vk = {V1, . . . , Vk} is the required one.

27.2. Remark (characterization of normality). A space X is normal if and

only if for any of its open covers u = {U1, . . . , Uk} there exists an

open cover v = {V1, . . . , Vk} of the space X such that Cl(Vi) ⊂ Ui

, i = 1, . . . , k.

Necessity was proved in Lemma 27.1. To prove sufficiency, consider disjoint closed sets F and T in X. The family of sets {UT = X \ F, UF = X \ T} —

is an open cover of X. There exists an open cover {VT , VF } of the space X such that

T ⊂ VT ⊂ Cl(VT ) ⊂ UT ,

F ⊂ VF ⊂ Cl(VF ) ⊂ UF .

Then F ⊂ OF = X \ Cl(VT ), T ⊂ OT = X \ Cl(VF ) and

OF ∩ OT = (X \ Cl(VF ))\

(X \ Cl(VT )) = X \ (Cl(VF )

[

Cl(VT )) = ∅.

27.3. Definition. A space X is called compact if from any of its open

covers a finite subcover can be chosen.

27.4. Examples. 1. Finite spaces. Spaces whose topology is finite.

2. A segment (Borel–Lebesgue Lemma). Closed bounded subsets of R

3. A space with the finite complement topology. The line in the Zariski topology.

27.5. Definition. A space X is called finally compact if from any of

its open covers a countable subcover can be selected (hereafter countability

is understood as at most countable).

27.6. Examples. 1. Countable spaces. Spaces that are a countable union of compact subspaces.

2. Spaces satisfying the second axiom of countability. Separable metrizable

spaces. Spaces R

3. The Sorgenfrey line.

Proof (of the final compactness of the Sorgenfrey line). Let u — be an arbitrary

open cover of the Sorgenfrey line X, and let VO — be the interior of the set O ∈ u with respect to the usual (Euclidean) topology of the line. Let us show that the set A = X\

{VO : O ∈ u}

— is countable.

For any x ∈ A there exist O ∈ u and ax ∈ X, x < ax, such that [x, ax) ⊂ O. Thus,

if x 6= x

, x, x0 ∈ A, then [x, ax) ∩ [x

, ax0 ) = ∅ (otherwise either x

0 ∈ (x, ax), or

x ∈ (x

, ax0 ), but the intervals (x, ax) and (x

, ax0 ) contain no points of A). Since the cardinality of pairwise

disjoint half-intervals on the line is at most countable, |A| ≤ ℵ0.

The set R\A =

{VO : O ∈ u} — is a separable metrizable space. Therefore from its

open cover {VO : O ∈ u} a countable subcover can be chosen. Since VO ⊂ O, then from

u a countable subfamily u

can be chosen, the union of whose elements contains X \ A. From

u a countable subfamily u

00 can be chosen, the union of whose elements contains A. Then

0 ∪ u

00 — is a countable subcover of u.

27.7. Definition. A cover u is inscribed in a cover v if every element U ∈ u is contained in some set V ∈ v.

27.8. Remark. A space X is compact (resp. finally compact) if and only

if into any of its open covers a finite (resp. countable) open cover

can be inscribed.

27.9. Definition. A family u of subsets of a space X is called locally finite if for every point x ∈ X there exists a neighborhood Ox intersecting only finitely

many elements of the family u.

Any subfamily of a locally finite family of sets is a locally finite

family of sets.

27.10. Definition. A space X is called paracompact if into any of its open cover a locally finite open cover can be inscribed.

27.11. Proposition. The properties of compactness, final compactness, and paracompactness are inherited when passing to closed subspaces.

We say that a family of sets u is inscribed in a family of sets v if every

element U ∈ u is contained in some set V ∈ v.

27.12. Lemma. Let F — be a closed subset of a paracompact space X

and let u be a family of open subsets of the space X such that F ⊂

{U ∈ u}.

Then there exists a locally finite family u0 of open subsets of X such that u0

is inscribed in u and F ⊂

{U ∈ u0}.

Proof. The family u ∪ {X \ F} — is an open cover of X. Let us inscribe into it a locally

finite cover v. The family u0 = {V ∈ v : V ∩ F 6= ∅} is the required one.

27.13. Definition. A family u of subsets of a space X is called conservative if

Cl[

U ∈ u0

[

Cl(U) : U ∈ u0

(27.1)

for every subfamily u0 ⊂ u.

27.14. Proposition. Every locally finite family is conservative.

Proof. Equality (27.1) is composed of two inclusions ⊃ and ⊂. The inclusion ⊃

follows from the monotonicity of the closure operation. Let us check that the inclusion ⊂ holds. Let

x ∈ ClS

{U ∈ u0}

. There exists a neighborhood Ox intersecting a finite subfamily of elements of u0, namely, U1, . . . , Uk. Then x ∈ Cl(U1 ∪ . . . ∪ Uk) = Cl(U1) ∪ . . . ∪ Cl(Uk) ⊂

Cl(U) :

U ∈ u0

27.15. Theorem. Every Hausdorff paracompact space X is normal.

Proof. First let us show that X is regular. Since X is Hausdorff, we need to check that axiom T3 holds. Let a closed set F ⊂ X and a point x0 ∈ X\F be given. For

every point x ∈ F there exist disjoint neighborhoods Oxx0 and Ox of the points x0 and x respectively. Note that x0 6∈ Cl(Ox). Into the family

Ox : x ∈ F}, for which F ⊂

Ox : x ∈ F

by Lemma 27.12, we can inscribe a locally finite family v, of sets open in X,

such that F ⊂

{V ∈ v}.

But v — is locally finite and, by Proposition 27.14, is a conservative family of sets. Therefore

Cl[

{V ∈ v}) = [

{Cl(V ) : V ∈ v} ⊂ X \ {x0},

and Ox0 = X \ ClS

{V ∈ v}) is a neighborhood of the point x0. The set OF =

{V ∈ v} is a neighborhood of the set F and, obviously, Ox0 ∩ OF = ∅. Regularity of the space X is proved.

Replacing the pair (F, x0) with a pair of disjoint closed sets (F1, F2) and applying an analogous procedure, we prove the normality of a regular paracompact space X.

27.16. Corollary. Every Hausdorff compact space X is a Hausdorff paracompact space (hence normal).

27.17. Theorem. Every regular finally compact space X is a Hausdorff paracompact space (hence normal).

Proof. Let u — be an open cover of X. For an arbitrary set U ∈ u and

a point x ∈ U, by the regularity of X there exists a neighborhood OU x of the point x such that

U x

⊂ U. (27.2)

By the final compactness of X, from the open cover

Ω =

U x : U ∈ u, x ∈ X

(27.3)

a countable subcover can be selected

Ω1 =

Uixi

: i ∈ N

. (27.4)

From (27.2) and (27.4) it follows that the family

u0 = {Ui

, i ∈ N} (27.5)

is a cover of the space X. Let us define the sets Vi

, i ∈ N, as follows:

V1 = U1, Vj+1 = Uj+1 \

[

Uixi

: i ≤ j

. (27.6)

Let us show that v = {Vj : j ∈ N} is an open locally finite cover, inscribed

in the cover u0 and, consequently, in the cover u.

Let us begin by noting that according to (27.6) the family v consists of open sets and is inscribed in

the cover u0.

Let us prove that v — is an open cover of X. Take an arbitrary point x ∈ X. There exists

a smallest such j that x ∈ Uj . Then x /∈ Cl

OUixi

for i < j and, consequently, x ∈ Vj according to

(27.6).

It remains to check the local finiteness of v. Let x ∈ X. Since Ω1 — is a cover, x belongs to some set OUixi

. Then from (27.6) it follows that OUixi is a neighborhood of the point x not intersecting the set Vj for j > i. Hence, the cover v is locally

finite.

Regular finally compact spaces are called Lindelöf spaces.

27.18. Theorem (A. Stone). Every metrizable space is paracompact.

27.19. Example. The Sorgenfrey line is an example of a paracompact (finally compact) space whose square is not paracompact (not finally compact).

Assignment No. 8

1. Prove that two metrics on the same set are topologically equivalent

if and only if every convergent sequence of points of this set in

one metric also converges in the other.

2. Let two metrics ρ1 and ρ2 be given on a set X. Prove that if there exist real numbers k1 > 0 and k2 > 0 such that ρ1(x, y) ≤ k2ρ2(x, y) and ρ2(x, y) ≤ k1ρ1(x, y) for

any x, y ∈ X, then the metrics ρ1 and ρ2 are topologically equivalent.

Suppose the metrics ρ1 and ρ2 on X are topologically equivalent. Do they satisfy the condition of the problem?

3. Prove that the topology of the Hilbert space `

is stronger than the topology on `

2 as a

subspace of the Tychonoff product R

ℵ0

4. Prove that the Tychonoff product R

ℵ0 — is a linear topological space (i.e.

the naturally defined operations of addition and scalar multiplication are continuous).

Prove that any subspace of the Tychonoff product R

ℵ0