91Ó°ÊÓ

Let \(S\) be a set. A relation \(a \preceq b\) defined for certain pairs in \(S\) is called a partial order on \(S\) if it satisfies the following axioms. (a) \(a \preceq a\) for all \(a \in S\). (b) If \(a \preceq b\) and \(b \preceq a\), then \(a=b\). (c) If \(a \preceq b\) and \(b \preceq c\), then \(a \preceq c\). We then say that \(S\) is partially ordered by \(\preceq\). Show that the set \(\mathcal{C}[0,1]\) of continuous functions on \([0,1]\) is partially ordered by the relation \(f \preceq g\) if \(f(x) \leq g(x)\) for all \(x \in[0,1]\).

Let \(\ell_{\infty}\) denote the set of all bounded sequences of real numbers, let $$ x=\left\\{x_{1}, x_{2}, \ldots\right\\} \text { and } y=\left\\{y_{1}, y_{2} \ldots\right\\} $$ belong to \(\ell_{\infty}\), and let $$ d_{\infty}(x, y)=\sup _{k}\left|x_{k}-y_{k}\right| . $$ Show that \(d_{\infty}\) is a metric on \(\ell_{1}\).

Let \(\left\\{x_{1}, x_{2}, \ldots, x_{m}\right\\}\) be a finite set of points in a metric space \((X, d)\). Show that $$ d\left(x_{1}, x_{m}\right) \leq \sum_{i=1}^{m-1} d\left(x_{i}, x_{i+1}\right) $$

What subsets of the space \((X, d)\), where \(d\) is the discrete metric, are compact?

Show that the set of polygonal functions on \([a, b]\) with rational vertices is a countable dense subset of \(\mathcal{C}[a, b]\).

In a general metric space \((X, d)\) take an arbitrary set \(A \subset X\) and perform a sequence of complements or closures. How many distinct sets can arise in this way?

Let \(E\) be a closed set in a metric space \((X, d)\) and let \(x\) be a point that is not in \(E\). Show that $$ \inf \\{d(x, y): y \in E\\}>0 $$ Show that if \(E\) and \(F\) are disjoint closed sets in a metric space, then it is not necessarily true that $$ \inf \\{d(x, y): x \in E, y \in F\\}>0 $$

Let \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) be Cauchy sequences in a metric space \((X, d)\). Show that \(d\left(x_{n}, y_{n}\right)\) converges even if the sequences \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) themselves do not.

Show that $$ Y=\\{f \in M[a, b]: f \text { is one-to-one }\\} $$ is a residual subset of the space \(M[a, b]\) of bounded functions on \([a, b]\).

Is it true in an arbitrary metric space that every finite set is nowhere dense?