91Ó°ÊÓ

Prove that the real line \(R\) is not compact.

Let \(X\) be a compact space and \(\left(F_{n}\right)_{n}=1,2,3, \ldots\) a sequence of nonempty closed subsets of \(X\) such that \(F_{n+1} \subset F_{n}\) for each \(n\). Prove that \(\bigcap_{n=1}^{\infty} F_{n} \neq \emptyset .\)

Prove that every finite subset of a topological space is compact.

Let \(X\) and \(Y\) be topological spaces satisfying the second axiom of countability. Prove that \(X \times Y\) also satisfies the second axiom of countability and hence \(R^{n}\) does.

Let \(X\) be a topological space. A family \(\left\\{F_{\alpha}\right\\}_{\alpha \in I}\) of subsets of \(X\) is said to have the finite intersection property if for each finite subset \(J\) of \(I\), \(\cap_{\alpha \in J} F_{\alpha} \neq \varnothing\). Prove that \(X\) is compact if and only if for each family \(\left\\{F_{\alpha}\right\\}_{\alpha \in I}\) of closed subsets of \(X\) that has the finite intersection property, we have \(\cap_{\alpha \in J} F_{\alpha} \neq \varnothing\).

A subset \(A\) of a topological space \(X\) is called dense if \(\bar{A}=X\). Let \(X\) be a compact metric space. Prove that there is a sequence \(a_{1}, a_{2}, \ldots\) of points of \(X\) such that the set \(A=\left\\{a_{1}, a_{2}, \ldots\right\\}\) is dense in \(X\).