91Ó°ÊÓ

Let \(B \subset \mathbb{R}^{n}\) be any convex set, \(X\) any topological space, and \(A\) any subset of \(X\). Show that any two continuous maps \(f, g: X \rightarrow B\) that agree on \(A\) are homotopic relative to \(A\).

Let \(X\) be a path connected space and \(p, q \in X\). Determine an algebraic necessary and sufficient condition on the fundamental group of \(X\) under which all path classes from \(p\) to \(q\) give the same isomorphism of \(\pi_{1}(X, p)\) with \(\pi_{1}(X, q)\).

For any compact surface \(S\), show that \(S\) with one point removed is homotopy equivalent to a bouquet of finitely many circles.

Given any collection \(\left\\{X_{\alpha}: \alpha \in A\right\\}\) of topological spaces, show that their disjoint union \(\mathrm{I}_{\alpha} X_{\alpha}\), together with the disjoint union topology and the natural inclusions \(\iota_{\alpha}: X_{\alpha} \hookrightarrow \mathrm{L}_{\alpha} X_{\alpha}\), is their sum in the category TOP.

Given any collection \(\left\\{X_{\alpha}: \alpha \in A\right\\}\) of topological spaces, show that their disjoint union \(\mathrm{I}_{\alpha} X_{\alpha}\), together with the disjoint union topology and the natural inclusions \(\iota_{\alpha}: X_{\alpha} \hookrightarrow \mathrm{L}_{\alpha} X_{\alpha}\), is their sum in the category TOP.

Given any collection \(\left\\{X_{\alpha}: \alpha \in A\right\\}\) of topological spaces, define a basis in the Cartesian product \(\prod_{\alpha} X_{\alpha}\) consisting of product sets of the form \(\prod_{\alpha} U_{\alpha}\), where \(U_{\alpha}\) is open in \(X_{\alpha}\) and \(U_{\alpha}=X_{\alpha}\) for all but finitely many \(\alpha\). Show that this is a basis, and that \(\prod_{\alpha} X_{\alpha}\) with this topology is the product of the spaces \(X_{\alpha}\) in the category TOP.

Show that any vertex in a connected finite tree is a strong deformation retract of the tree.