Download MATH 730: PROBLEM SET 2 (1) (a) Let X be a locally compact

MATH 730: PROBLEM SET 2
WILLIAM GOLDMAN
(1) (a) Let X be a locally compact Hausdorff space. Then the
intersection of an open subset of X and a closed subset of
X is locally compact.
(b) Let X be Hausdorff and Y ⊂ X be locally compact. Show
that Y is the intersection of an open subset of X and a
closed subset of X.
(2) Prove or disprove: The continuous image of a locally compact
space is locally compact.
(3) (Bredon §I.11.1, p.31) Recall that the quasicomponents of a
topological space X are the equivalence classes under the equivalence relation
f
x ∼ y :⇐⇒ f (x) = f (y)∀X →
− D where D is a discrete space.
Prove or disprove:
(a) If X is compact and Hausdorff, then the quasi-components
are the connected components of X.a
(b) Same if X is only assumed to be compact.
(c) Same if X is only assumed to be Hausdorff.
(4) Find an example of a continuous open map which is not cloaed,
and a continuous closed map which is not open. Must every
continuous map be either open or closed?
(5) Prove or disprove: If a topological space X satisfies X ≈ X ×X,
then X = ∅.
Date: 14 September 2006.
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