Problem 1

Exercise 8.1 (Types of compactness)   Give the definitions of compactness and limit point compactness of a topological space. Show that every compact space is limit point compact. Give an example that the converse is not true.

Definition 1 (Compactness)   A topological space $X$ is called compact if for every open cover of $X$, there exists a finite subcollection of that cover which also covers $X$.

Definition 2 (Limit point compactness)   A topological space $X$ is called limit point compact every infinite subset $S\subseteq X$ has a limit point.

Now for the real workout:

Proof. If $X$ is compact, then $X$ is limit point compact. Suppose not, then for any $x\in X$, select an open set $U\ni x$ such that $S\cap U \subseteq \{x\}$. Cover

$$X \subseteq \bigcup_{x\in X} U$$ (8.1)

$$\subseteq \bigcup_{i=1}^n U_i$$ (8.2)

The infinite set $S$ can be included

$$S = \bigcup_{i=1}^n U_i\cap S \subseteq \bigcup_{i=1}^n \{x\}$$ (8.3)

which contradicts that $S$ is an infinite subset. Therefore, $S$ is limit point compact.

To see a limit point compact space which is not compact, consider $\mathbb{Z}\times\{0,1\}$ where $\mathbb{Z}$ has the standard topology and the topology of $\{0,1\}$ is $\mathcal{T} = \{\{\}, \{0,1\}\}$. Any point is a limit point, so any infinite subset contains a limit point. $\qedsymbol$