Master’s Exam Topology June 1993

1. (a) Give the definition of a regular topological space.

(b) Prove that every subspace of a regular space is regular.

2. Let X_{α} be a collection of spaces indexed by α ∈ J. Let α_{0} ∈ J. Prove that the projection

map

π_{α_{0}} : Π_{α∈J}X_{α} → X_{α_{0}}

is a quotient map.

3. Let X be a topological space. A subspace A ⊆ X is locally closed if any x ∈ A has an open neighborhood U in X such that U ∩ A is closed in U (though U ∩ A is not necessarily closed in X). Prove that A is locally closed if and only if there is an open set V ⊆ X and a closed set C ⊆ X such that A = V ∩ C.

4. Prove the tube lemma: consider the product space X × Y, where Y is compact. Let x_{0} ∈ X. Let N be an open set of X × Y containing the slice {x_{0}} × Y. Then there is an open set W ⊆ X containing x_{0} such that N contains the tube W × Y.