91Ó°ÊÓ

Let \(X\) be a totally ordered set (see the Appendix), and assume that \(X\) has at least two elements. For any \(a \in X\), define sets \(L(a), R(a) \subset X\) by $$ \begin{aligned} &L(a)=\\{c \in X: ca\\} \end{aligned} $$ Give \(X\) the topology generated by the subbasis \(\\{L(a), R(a): a \in X\\}\), called the order topology. (a) Show that each set of the form \((a, b)\) is open in \(X\) and each set of the form \([a, b]\) is closed (where \((a, b)\) and \([a, b]\) are defined just as in \(\mathbb{R}\) ). (b) Show that \(X\) is Hausdorff. (c) Show that for any \(a, b \in X, \overline{(a, b)} \subset[a, b]\). Under what conditions does equality hold?

Let \(X\) be a second countable topological space. Show that every collection of disjoint open subsets of \(X\) is countable.

(a) Show that every second countable space has a countable dense subset. (b) Show that a metric space is second countable if and only if it has a countable dense subset.

Let \(X\) be a first countable space. (a) For any set \(A \subset X\) and any point \(p \in X\), show that \(p \in \bar{A}\) if an only if there is a sequence \(\left\\{p_{n}\right\\}_{n=1}^{\infty}\) in \(A\) such that \(p_{n} \rightarrow p\). (b) Show that for any space \(Y\), a map \(f: X \rightarrow Y\) is continuou if and only if \(f\) takes convergent sequences in \(X\) to converger sequences in \(Y\).

Show that any manifold has a basis of Euclidean balls.

Suppose \(M\) is an \(n\)-dimensional manifold with boundary. Show that Int \(M\) is an \(n\)-manifold and \(\partial M\) is an \((n-1)\)-manifold (without boundary).