91Ó°ÊÓ

. In the real line, prove that the boundary of the open interval \((a, b)\) and the boundary of the closed interval \([a, b]\) is \(\\{a, b\\}\).

Let \(f:(X, 3) \rightarrow\left(Y, 3^{\prime}\right)\) be a homeomorphism. Let a third topological space \(\left(Z, J^{\prime \prime}\right)\) and a function \(h:\left(Y, J^{\prime}\right) \rightarrow\left(Z, J^{\prime \prime}\right)\) be given. Prove that \(h\) is continuous if and only if \(h f\) is continuous. Let another function \(k:\left(Z, J^{\prime \prime}\right) \rightarrow(X, J)\) be given. Prove that \(k\) is continuous if and only if \(f k\) is continuous.

Let \((X, J)\) be a topological space. Prove that \(\varnothing, X\) are closed sets, that a finite union of closed sets is a closed set, and that an arbitrary intersection of closed sets is a closed set.

Prove that a subspace of a metrizable space is a metrizable space.

The "rational density theorem" for the real line states that between any two real numbers there lies a rational number. Use the rational density theorem to prove that the rational numbers are dense in the real line.

The "Archimedean principle" for the real line states that if \(c, d>0\) then there is a positive integer \(N\) such that \(N c>d\). Prove the Archimedean principle for the real line and use this principle to prove the rational density theorem for the real line.