Download Homework 3 Topology I, Fall 2014

Homework 3
Topology I, Fall 2014
Problem 1. (a) Show that a subspace of a completely regular space is completely regular.
(b) Show that completely regular space is regular.
(c) Show that the product of completely regular spaces is completely regular.
Problem 2. Let X be a compact Hausdorff space. Show that X is metrizable if and only if it is
second countable.
Problem 3. Let X be a locally compact space and let A be a subset of X such that, for every
compact subset K of X, the intersection A ∩ K is a closed subset of X. Show that A is a closed
subset of X.
Problem 4. Let X be a locally compact space and A a closed subspace of X. Show that A is locally
compact.
(Be careful, since we do not assume that X is Hausdorff, we cannot assume that compact
subspaces are closed.)
Problem 5. Show that a product of a paracompact space and a compact space is paracompact.
Problem 6. Let X be a topological space and {Aj | j ∈ J} a locally finite family of sets in X.
Show that {Aj | j ∈ J} is locally finite.
Problem 7. Consider the space = R/ ∼, where x ∼ y ⇐⇒ x − y is rational. Show that X is an
uncountable space with trivial topology.
Problem 8. Consider the equivalence relation ∼ on I = [0, 1] given by
x ∼ y ⇐⇒ x = y or 1/3 < x, y < 2/3,
and the quotient space X = I/ ∼. Prove or disprove each of the following
(a) X is Hausdorff.
(b) X is connected.
(c) X is compact.