Lecture
31.3. Theorem. Let X — be a Tychonoff space. There exists a compactification
(Y, i) of the space X such that any bounded continuous function f : X → R has
a unique extension ˜f : Y → R (i.e. f = ˜f ◦ i).
Proof. Let C
∗
(X) = {fα : X → R : α ∈ A} — be the set of all bounded
continuous functions on X. We may assume that fα(X) ⊂ [mα, Mα], α ∈ A. Then the mapping
i = ∆α∈Afα : X →
Y
α∈A
[mα, Mα]
is an embedding (Lemma 25.3 of Lecture 7) into a compact space (Theorem 30.2). Set
Y = Cl(i(X)) (the closure of i(X) in the product Q
α∈A[mα, Mα]).
Then (Y, i) — is a compactification of X. Let f ∈ C
∗
(X). Since f = fα for some α ∈ A,
then f = prα ◦ i (see Definition 17.7 of Lecture 4), and the mapping prα|Y : Y → [mα, Mα] is
the required extension of f. Uniqueness of the extension follows from Proposition 22.4 of Lecture
5.
The compactification constructed in Theorem 31.3 is called the Stone-Čech compactification and is denoted βX.
31.4. Problem. Prove its uniqueness (up to equivalence).
31.5. Example. The square of the Sorgenfrey line is a Tychonoff space. It has a compactification which is normal. Thus, we obtain an example of a normal space
having an everywhere dense non-normal subspace.
§ 32. Local compactness. The Alexandroff one-point compactification.
32.1. Definition. A space X is called locally compact if for an arbitrary point x and its neighborhood Ox there exists a neighborhood Ux such that Cl(Ux) ⊂ Ox and
the space Cl(Ux) is compact.
32.2. Examples. 1. Euclidean space R
n, n ∈ N, and topological manifolds are locally compact spaces.
2. An open (closed) subset of a locally compact space is locally compact.
3. The line in the Zariski topology — is a compact, non-locally compact space.
4. A Hausdorff compact space is locally compact.
32.3. Theorem. Let X — be a Hausdorff space. Then X — is locally compact if and
only if
either X — is a compact space,
or there exists a compactification Y of the space X such that Y \ X is a single point.
If X — is a locally compact, non-compact space, then any two of its one-point compactifications are equivalent (called the Alexandroff one-point compactification,
notation αX).
Proof. Let us prove the uniqueness of the one-point compactification. Let (Yj , ij ),
where Yj \ ij (X) = {yj}, j = 1, 2, — be compactifications of X. Set f : Y1 → Y2, f|i1(X) = i2 ◦ i
−1
1
,
f(y1) = y2. The mapping f — is a bijection, moreover, f|i1(X)
: i1(X) → i2(X) — is a homeomorphism.
It remains to check the continuity of f. If O is open in Y2, then either y2 6∈ O, or y2 ∈ O. In the first
case O is open in i2(X). Hence f
−1
(O) is open in i1(X), and open in Y1 (since i1(X)
is open in Y1). In the second case Y2 \ O — is a compact subset of i2(X). Then f
−1
(Y2 \ O)
— is a compact subset of i1(X) (and hence of Y1), which is closed in Y1. Thus, f
−1
(O)
is open in Y1 and f is continuous. Hence f — is a homeomorphism such that f ◦ i1 = i2.
Proof of the first statement. Necessity. If X is not compact, then we define
a topology on the set Y = X ∪ {y} (y 6∈ X). The open sets are the open
subsets of X and sets of the form Y \ K, where K — is a compact subset of X. It is easy to check: the correctness of the definition of the topology and that X — is an everywhere dense subspace
Y .
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Hausdorffness of Y . Consider the case of points x ∈ X and y. Since X is locally compact, there
exists a neighborhood Ox whose closure Cl(Ox) is compact. Then Ox and Y \ Cl(Ox) —
are disjoint neighborhoods of the points x ∈ X and y respectively.
Compactness of Y . Let u ∈ cov(Y ) and y ∈ U ∈ u. Then K = Y \U — is a compact subset
of X, hence there exists a finite subfamily u
0 ⊂ u covering K, and {U} ∪ u
0 — is the required
finite subcover of u.
Sufficiency follows from items 2 and 4 of Example 32.2.
32.4. Corollary. A space X is Hausdorff and locally compact if and
only if X is homeomorphic to an open subset of a Hausdorff compact
space.
A Hausdorff locally compact space X is Tychonoff.
32.5. Example. The one-point compactification of R
n is the sphere S
n, n ∈ N. In the one-dimensional case the circle S
1
, in the two-dimensional case (if R
2 is regarded as the complex
numbers) the sphere S
2 is called the closed complex plane or the Riemann sphere.
§ 33. Metrizable compact spaces. Characterizations of compactness of metrizable spaces.
33.1. Theorem. A metrizable compact space X satisfies the second axiom
of countability.
Proof. Let ρ — be a metric on the space X. From the cover of X by open balls O 1
n
(x) one can choose a finite subcover O 1
n
(x
n
1
), . . . , O 1
n
(x
n
k(n)
), n ∈ N. Then the countable
set D =
S∞
n=1{x
n
1
, . . . , xn
k(n)
} is everywhere dense in Y . Indeed, if O is a nonempty open
subset of X, then there exists an open ball Oε(x) ⊂ O. For any open ball O 1
n
(x
n
i
),
where 1
n < ε and x ∈ O 1
n
(x
n
i
) we have x
n
i ∈ O(x) ⊂ O. By Theorem 21.7 of Lecture 5 X satisfies the second
axiom of countability.
33.2. Corollary. Let X — be a metrizable space. Then X — is separable if and
only if there exists a metrizable compactification of X (not necessarily
unique).
Proof. Necessity follows from Theorem 21.7 of Lecture 5 (there exists a countable base of
X), from the proof of Urysohn's Theorem 26.6 of Lecture 7 (embedding X in the Hilbert cube), from Tychonoff's Theorem
30.2 of Lecture 9 (compactness of the Hilbert cube), and from Proposition 27.11 (compactness
of the closure of the image of X in the Hilbert cube). Sufficiency follows from Theorem 33.1 and Theorem
21.3 of Lecture 5 (countable base on X).
33.3. Theorem. Let X — be a metrizable space. Then the following conditions are equivalent:
(a) X — is compact;
(b) any sequence of points in X contains a convergent subsequence
(sequential compactness of X);
(c) any continuous function f : X → R is bounded (pseudocompactness of X).
Proof. Fix a metric ρ on the space X that generates its topology.
(a) ⇒ (b) Let (xn) — be a sequence of points in X. We may assume that all points of the sequence are pairwise distinct (otherwise either there exists a subsequence of
pairwise distinct points, and one can consider it, or there exists a subsequence,
all of whose points coincide and which is obviously convergent). If we show that the set
A = {xn : n ∈ N} has a limit point x
∗
, then, taking a point xin ∈ O 1
n
(x
∗
), in+1 > in,
n ∈ N, we obtain a subsequence (xin
) converging to x
∗.
Suppose that A has no limit points. Then for every point x ∈ X there exists
a neighborhood Ox containing only finitely many points of A. By the compactness of X, from the cover
{Ox : x ∈ X}, one can choose a finite subcover. It follows that the set A is finite,
which contradicts the assumption about it.
(b) ⇒ (c) If f : X → R — is an unbounded continuous function (we may assume, without loss of generality, that f is unbounded above), then there exists a sequence of points (xn)
such that f(xn+1) ≥ f(xn) + 1, n ∈ N. The sequence (xn) contains, converging to the point
x
∗ a subsequence (xin
). But then by Definition 13.7 of Lecture 3 (continuity in the
57
Heine sense) the function f is discontinuous at the point x
∗
. The resulting contradiction completes the proof
of the implication.
(c) ⇒ (a) For every ε > 0 there exists a finite set of points Aε = {x1, . . . , xn} such that for any point x ∈ X there exists a point xi ∈ Aε, for which ρ(x, xi) < ε (the set
of points {x1, . . . , xn} is called a finite ε-net). Indeed, otherwise one could construct a sequence of points (xn) with pairwise distances ≥ ε. The set
A = {xn : n ∈ N} would be a closed subset of X and a discrete space (an open
ball Oε
2
(x), x ∈ X, can contain at most one point of the set A), on which the function
f : A → N, f(xn) = n, n ∈ N, is continuous. By the Brouwer–Tietze–Urysohn Theorem (Corollary 23.4
of Lecture 6) there exists a continuous extension of f to X, which is unbounded. The resulting
contradiction completes the proof of the existence of a finite ε-net.
The union A =
S∞
n=1 A 1
n
of finite 1
n
-nets, n ∈ N, is a countable everywhere dense subset of X. Since X is separable, by Corollary 33.2 there exists a metrizable compactification Y of the space X, and let ρ
0 — be a metric on Y generating its topology. If
there exists a point y ∈ Y \ X, then it is not isolated, hence the function f : X → R,
f(x) = 1
ρ0(x,y) — is a continuous unbounded well-defined function on X, which contradicts condition (c). Hence X = Y and X — is a compact space.
Assignment No. 10
1. Prove that any mapping f : X → Y of a Tychonoff space X into a compact
space Y extends to a mapping of its Stone-Čech compactification βX.
Establish the uniqueness (up to equivalence) of the Stone-Čech compactification βX of a Tychonoff space X.
2. Give an example of a Tychonoff space having no metrizable compactification.
3. Prove. a) Euclidean spaces R
n, n ∈ N, are locally compact spaces.
b) An open (closed) subset of a locally compact space is locally compact.
c) The line in the Zariski topology — is a compact, non-locally compact space.
d) A Hausdorff compact space is locally compact.
4. Prove that Q is not a locally compact space.
5. In which of the topologies from Example 8.5 of Lecture 2 is the line locally compact?
6. Prove that the Alexandroff one-point compactification of R
n — is the sphere S
n, n ∈ N.
7. Prove that the Alexandroff one-point compactification of the countable discrete space N — is the convergent sequence {0} ∪ (
1
n
).
8. Prove that a homeomorphism of Tychonoff spaces extends to a homeomorphism
of their Stone-Čech compactifications.
Prove that a homeomorphism of Hausdorff locally compact spaces extends to a homeomorphism of their Alexandroff one-point compactifications.
9. Is the continuous Hausdorff image of a locally compact space locally
compact?
10. Prove that any non-compact metrizable space contains an infinite closed discrete subspace.
11. Prove that on any non-compact metrizable space X there exists a continuous function f : X → (0, 1], for which inf{f(x) : x ∈ X} = 0.
Additional Problems for Assignment No. 10
12. For compactifications (Y1, i1) and (Y2, i2) we say Y1 ≤ Y2 if there exists a mapping
f : Y2 → Y1 such that f ◦ i2 = i1. Prove that f(Y2 \ i2(X)) = (Y1 \ i2(X)).
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Prove that ≤ — is an ordering on the set of compactifications, for which βX — is the greatest element. If the space X is locally compact, then αX — is the least element.
13. Prove that a Hausdorff space is locally compact if and only if,
it is an open subset of every one of its compactifications.
14 (Alternative definition of local compactness of a space). A space
X is called locally compact if for an arbitrary point x there exists a
neighborhood Ux such that Cl(Ux) is a compact subspace of X.
Will an open (closed) subset of a locally compact space be locally compact
in this sense?
Will the line in the Zariski topology be locally compact in this sense?
Prove that for Hausdorff spaces both notions of local compactness coincide.
15. Prove that any compact space is pseudocompact.
Prove that the set T(ω1) of countable transfinites (Example 7.1 of Lecture 1) is a locally compact, non-compact Hausdorff space which is pseudocompact and
sequentially compact.
Prove that βT(ω1) = αT(ω1) = T = T(ω1) ∪ {ω1}.
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