Download Local compactness - GMU Math 631 Spring 2011

Local compactness
These notes discuss the same topic as Section 29 of Munkres’ book.
Definition 1. A topological space X is called locally compact at a point x ∈ X if
there exists a neighborhood Ux 3 x such that Ux is compact. X is called locally
compact if X is compact at every point.
An equivalent definition: X is locally compact at x ∈ X iff there is a closed set
C ⊂ x such that C contains soe neighborhood of x.
Lemma 2. Let X and Y be topological spaces, g : X → Y a function, and A and
B two closed subspaces of X such that A ∪ B = X. If g|A and g|B are continuous
functions then g is a continuous function.
Proof. Let K be a closed subspace of Y . It suffices to show that g −1 (K) is closed
in X. We have g −1 (K) = g −1 (K ∩ g(X)) = g −1 (K ∩ g(A ∪ B))) = g −1 ((K ∩ g(A)) ∪
(K ∩ g(B)))) = g −1 (K ∩ g(A)) ∪ g −1 (K ∩ g(B)) = (g|A )−1 (K ∩ g(A)) ∪ (g|B )−1 (K ∩
g(B)). So g −1 (K) is closed as the union of two closed sets. 2
Proposition 3. A Hausdorff locally compact space is Tychonoff.
Proof. Let X be a Hausdorff locally compact space, x ∈ X, and F be a closed
subset of X such that x 6∈ F . There is an open set U such that x ∈ U and U is
compact. Put H = (F ∪ X \ U ) ∩ U . Then H is a closed subset of U and x 6∈ H.
U is Huasdorff and compact, hence Tychonoff. So there is a continuous function
f : U → I such that f (x) = 0 and f (H) ⊂ {1}. Define a function g : X → I by
f (x) if t ∈ U
g(t) =
1 otherwise.
Then g(x) = 0 and g(F ) ⊂ {1}. g is continuous by Lemma 3 (A and B in Lemma 3
are U and X \ U .) 2
Theorem 4. Let (X, T ) be a Hausdorff space. The following conditions are equivalent:
(1) X is locally compact;
(2) X is homeomorphic to an open subspace of some Hausdorff compact space;
(3) There exists a Hausdorff compact space Z such that X is an open dense
subspace of Z and |Z \ X| ≤ 1.
Proof. (1) ⇒ (3): If X is compact, then it suffices to put Z = X. So let X be
locally compact but not compact. Put K = {U : U ∈ T and U is compact}. Pick
p 6∈ X and put Z = X ∪ {p}. Further, put Bp = {{p} ∪ (X \ K) : K ∈ K} and
B = Bp ∪ T . The following facts are easy to check:
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•
•
•
•
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B is a base for a topology on Z.
Z (with the topology generated by B) is Hausdorff.
X is open in Z.
X is dense in Z.
Z (with the topology generated by B) is compact.
The topology inherited on X from Z coincides with the original topology
T.
1
2
(3) ⇒ (2) is obvious.
(2) ⇒ (1): Let X be an open subspace of a Hausdorff compact space Y . Let
x ∈ X. Being Hausdorff and compact, Y is regular. Since X is open in Y , there is
an open set V ⊂ Y such that x ∈ V ⊂ V ⊂ X. V is a closed subspace of a compact
space Y , so V is compact. 2
Remark 5. If X is Hausdorff and locally compact but non compact, then the space
Z from Theorem 4 is often called the Alexandroff one-point compactification of X
and denoted aX.
Examples 6. The following spaces are locally compact:
(1) All compact spaces;
(2) Discrete space of any cardinality;
(3) R;
(4) More generally, Rn for any finite n;
(5) More generally, any manyfold;1
(6) A Ψ-space;
(7) ω1 (with the order topology);
(8) Tychonoff plank;
(9) Every closed subspace of a locally compact space;
(10) Every open subspace of a Hausdorff locally compact space;
(11) Every finite product of locally compact spaces.
Examples 7. The following spaces are not locally compact:
(1) Q;2
(2) RN ;
(3) Cp (X) where X is any non-discrete Tychonoff space;
(4) Niemytzky plane;
(5) Zorgenfrey line;
(6) The metric hedgehog;
(7) The quotient hedgehog.3
1Let n ∈ N. A Hausdorff topological space X is called a manyfold of dimension n if every point
of X has a neighborhood homeomorphic to Rn .
2In general, if X can be embedded in a larger Hausdorff space Y so that both X and Y \ X
are dense in Y then X is not locally compact.
3This shows that a continuous image of a locally compact space does not have to be locally
compact. Indeed, the quotient hedgehog is a quotient image of the discrete sum of countably
many copies of the unite interval.