Projections along a compact factor and the tube lemma

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Problem 2.1.2 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.1 "Point Set Topology"

Suppose that $A$ and $B$ are compact subspaces of spaces $X$ and $Y$ respectively, and that $N$ is an open neighborhood of

A \times B \subset X \times Y.

Prove that there are open sets $U \subset X$ and $V \subset Y$ such that

A \times B \subset U \times V \subset N.

Problem 2.1.6 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.1 "Point Set Topology"

(a)(b)

Let $f : X \to Y$ be a continuous function from a space $X$ to a Hausdorff space $Y$ . Let $C$ be a closed subspace of $Y$ , and let $U$ be an open neighborhood of $f^{-1}(C)$ in $X$ .

(a)

Prove that if $X$ is compact then there is an open neighborhood $V$ of $C$ in $Y$ such that $f^{-1}(V) \subset U$ .

(b)

Give a counterexample to show that if $X$ is not compact, then there need not be such a neighborhood $V$ .

Problem 2.1.16 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.1 "Point Set Topology"

Let $X$ and $Y$ be topological spaces, with $Y$ compact. Let $p : X \times Y \to X$ be the usual projection onto the first factor. Show that $p$ is a closed map.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.