4. Problems

Problem M-1. Show that the two descriptions of a $T_1$ space are equivalent.

Problem M-2. Consider the metric space $X = \{\frac 12, \frac 23, \frac 34, \dots \} \cup \{1\} \subseteq \mathbb{R}$. Describe the open sets of $X$. Which sets are clopen (simultaneously closed and open)?

Problem M-3. Given a topological space $X$ with topology ${\cal T}$, a family ${\cal U}$ of sets of $X$ is a base for the topology if every member of ${\cal T}$ is a union of members of ${\cal U}$ (possibly infinitely many). If $X$ is the real unit interval with the usual topology, are the open intervals with rational endpoints a base for the topology of $X$?

Problem M-4. A Boolean topological space is a compact Hausdorff space in which the clopen sets form a base for the topology. (For definitions, see problems and .) Show that the only connected subsets of a Boolean topological space are singletons. (A subset is said to be connected if it is connected in the relative topology.)

Problem M-5. Show that if a compact Hausdorff space $X$ has the property that every connected subset is a singleton, then $X$ is Boolean.

Problem M-6. Let $\omega _ 1$ be the ``first uncountable ordinal'', an uncountable well ordered set for which all principal ideals are countable. Let $\omega _ 1 ^ +$ be $\omega _ 1$ with a top element added. Give $\omega _ 1 ^ +$ the topology for which the open intervals ( $\{x : a < x < b\}$ or $\{x : a < x\}$ or $\{x : x < b\}$) form a basis.

(a) Is $\omega _ 1 ^ +$ compact Hausdorff with this topology?

(b) Are sequences adequate to determine whether a subset is closed?