A countably compact space is a topological space in which every countable open cover has a finite subcover. Prove that a second countable space is countably compact $\Leftrightarrow$ it is compact.

Let $Y$ be a subspace of a topological space $X$. If $Z$ is a non-empty subset of $Y$, show that $Z$ is compact as a subspace of $Y \Leftrightarrow$ it is compact as a subspace of $X$.

Let $X$ be a topological space. If $\left{X_i\right}$ is a non-empty finite class of compact subspaces of $X$, show that $\cup_i X_i$ is also a compact subspace of $X$. If $\left{X_j\right}$ is a non-empty class of compact subspaces of $X$ each of which is closed, and if $\bigcap_j X_j$ is non-empty, show that $\cap_j X_j$ is also a compact subspace of $X$.

Let $X$ be a compact space. We know by Theorem A that every closed subspace of $X$ is compact. By considering Example 16-3, show that a compact subspace of $X$ need not be closed.

If $\left{A_i\right}$ and $\left{B_j\right}$ are two classes of sets such that $\left{A_i\right} \subseteq\left{B_j\right}$, show that $\cup_i A_i \subseteq \cup_j B_j$ and $\bigcap_j B_j \subseteq \bigcap_i A_i$.

The difference between two sets $A$ and $B$, denoted by $A-B$, is the set of all elements in $A$ and not in $B$; thus $A-B=A \cap B^{\prime}$. Show the following:
$$
\begin{aligned}
A-B= & A-(A \cap B)=(A \cup B)-B ; \
& (A-B)-C=A-(B \cup C) ; \
& A-(B-C)=(A-B) \cup(A \cap C) ; \
& (A \cup B)-C=(A-C) \cup(B-C) ; \
& A-(B \cup C)=(A-B) \cap(A-C) .
\end{aligned}
$$

The symmetric difference of two sets $A$ and $B$, denoted by $A \Delta B$, is defined by $A \Delta B=(A-B) \cup(B-A)$; it is thus the union of their differences in opposite orders. Show the following:
$$
\begin{gathered}
A \Delta(B \Delta C)=(A \Delta B) \Delta C ; \
A \Delta \emptyset=A ; A \Delta A=\emptyset ; \
A \Delta B=B \Delta A ; \
A \cap(B \Delta C)=(A \cap B) \Delta(A \cap C) .
\end{gathered}
$$

A ring of sets is a non-empty class $A$ of sets such that if $A$ and $B$ are in A, then $A \Delta B$ and $A \cap B$ are also in $A$. Show that $A$ must also contain the empty set, $A \cup B$, and $A-B$. Show that if a nonempty class of sets contains the union and difference of any pair of its sets, then it is a ring of sets. Show that a Boolean algebra of sets is a ring of sets.

Show that the topological space defined in Example 16-5 is not a $T_1$-space.

Show that the topological space defined in Example 16-4 is a $T_1$-space but not a Hausdorff space.

If $X$ is a $T_1$-space with at least two points, show that an open base which contains $X$ as a member remains an open base if $X$ is dropped.

Let $X$ be a topological space, $Y$ a Hausdorff space, and $A$ a subspace of $X$. Show that a continuous mapping of $A$ into $Y$ has at most one continuous extension to a mapping of $\bar{A}$ into $Y$. Problem 17-2 is a special case of this statement.