91Ó°ÊÓ

Prove that a subspace of a metrizable space is metrizable.

Suppose that \(\mathrm{S}\) is a subspace of \(\mathrm{X}\). Show that the inclusion map \(\mathrm{S} \rightarrow \mathrm{X}\) is continuous. Furthermore, show that \(\mathrm{S}\) has the weakest topology (i.e. the least number of open sets) such that the inclusion \(\mathrm{S} \rightarrow \mathrm{X}\) is continuous.

\(\mathrm{X}\) is a topological space, \(\mathrm{S}\) is a subset and i: \(\mathrm{S} \rightarrow \mathrm{X}\) denotes the inclusion map. The set \(\mathrm{S}\) is given a topology such that for every space \(Y\) and map \(f: Y \rightarrow S\) f: \(Y \rightarrow S\) is continuous \(e\) if: \(Y \cdot X\) is continuous. Prove that the topology on \(\mathrm{S}\) is the fopalegy induced hy the lopu. logy on X.

Let \(Y\) be a subspace of \(X\) and let \(A\) be a subset of \(Y\). Denote by \(\mathrm{Cl}_{\mathrm{X}}(\mathrm{A})\) the closure of \(\mathrm{A}\) in \(\mathrm{X}\) and by \(\mathrm{Cl}_{\mathrm{Y}}\) (A) the closure of \(\mathrm{A}\) in \(\mathrm{Y}\). Prove that \(\mathrm{Cl}_{\mathrm{Y}}(\mathrm{A}) \subseteq \mathrm{Cl}_{\mathrm{X}}\) (A). Show that in general \(\mathrm{Cl}_{Y}(\mathrm{~A}) \neq \mathrm{Cl}_{\mathrm{X}}(\mathrm{A})\).

Show that the subset \((a, b)\) of \(R\) with the induced topology is homeomorphic to \(\mathrm{R}\). (Hint: Use functions like \(\mathrm{x} \rightarrow \tan (\pi(\mathrm{cx}+\mathrm{d}))\) for suitable \(\mathrm{c}\) and \(\mathrm{d}\).)

Let \(X, Y\) be topological spaces and let \(S\) be a subspace of \(X\). Prove that if \(f: X \rightarrow Y\) is a continuous map then so is \(f \mid S: S \rightarrow f(S)\).

Show that the subspaces \((1, \infty),(0,1)\) of \(\mathrm{R}\) with the usual topology are homeomorphic. (Hint: \(\mathrm{x} \rightarrow 1 / \mathbf{x}\).)