91Ó°ÊÓ

Show that the unit circle \(S^{1}\) in \(\mathbb{R}^{2}\) and the unit interval \([0,1]\) both are (Hausdorff) compactifications of \(\mathbb{R} .\) Hint: Use the fact that \(\mathbb{R}\) is homeomorphic to the open interval ] 0,1 [ and (therefore also) homeomorphic to \(\mathrm{S}^{1} \backslash\\{(1,0)\\}\).

An order isomorphism between two ordered sets \((X, \leq)\) and \((Y, \leq)\) is a bijective map \(\varphi: X \rightarrow Y\) such that \(x_{1} \leq x_{2}\) iff \(\varphi\left(x_{1}\right) \leq \varphi\left(x_{2}\right) .\) A segment of a well-ordered set \((X, \leq)\) is a subset of \(X\) of the form \(\min \\{x\\}\) for some \(x\) in \(X\), or \(X\) itself (the improper segment). Show that if \(X\) and \(Y\) are well-ordered sets, then either \(X\) is order isomorphic to a segment of \(Y\) (with the relative order) or \(Y\) is order isomorphic to a segment of \(X\). Hint: The system of order isomorphisms \(\varphi: X_{\varphi} \rightarrow Y_{\varphi}\) where \(X_{\varphi}\) and \(Y_{\varphi}\) are segments of \(X\) and \(Y\), respectively, is inductively ordered if we define \(\varphi \leq \psi\) to mean \(X_{\varphi} \subset X_{\psi}\) (which implies that \(\varphi=\psi \mid X_{\varphi}\) and thus \(Y_{\varphi} \subset Y_{\psi}\) ). Prove that for a maximal element \(\varphi: X_{\varphi} \rightarrow Y_{\varphi}\) either \(X_{\varphi}\) or \(Y_{\varphi}\) must be an improper segment.

Let \(\left(x_{\lambda}\right)_{\lambda \in \Lambda}\) be a net in a set \(X\). Show that the system \(\mathscr{F}\) of subsets \(A\) of \(X\), such that the net is in \(A\) eventually, is a filter (E 1.3.4). Show that the net converges to a point \(x\) in \(X\) iff the filter converges to \(x\).

(Connected spaces.) A topological space \((X, \tau)\) is connected if it cannot be decomposed as a union of two nonempty disjoint open sets. A subset of \(X\) is clopen if it is both open and closed. Show that \(X\) is connected iff \(\emptyset\) and \(X\) are the only clopen subsets. Let \(f: X \rightarrow Y\) be a surjective continuous map between topological spaces. Show that \(Y\) is connected if \(X\) is.

(Arcwise connected spaces.) A topological space \((X, \tau)\) is arcwise connected if for every pair \(x, y\) in \(X\) there is a continuous function \(f:[0,1] \rightarrow X\) such that \(f(0)=x\) and \(f(1)=y\). Geometrically speaking, \(f([0,1])\) is the curve or arc that joins \(x\) to \(y\). Show that an arcwise connected space is connected (E 1.4.13). Show that \(Y\) is arcwise connected if it is the continuous image of an arcwise connected space (cf. E 1.4.13).