Exercise 2, § 4

General Topology, I, 125

Let $X, Y$ be two topological spaces, $A$ a closed subset of $X \times Y$ . If $\mathfrak{B}(x)$ is the set of all neighbourhoods of a point $x \in X$ , show that

\bigcap_{V \in \mathfrak{B}(x)} \overline{A(V)} = A(x).

Give an example of an open subset

B

of a product

X \times Y

such that

\bigcap_{V \in \mathfrak{B}(x)} \overline{B(V)}

is different from

\overline{B_{(x)}}

Solution

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