What are closed and discrete sets in general topology?

Question

I have some idea about closed sets as well as discrete sets in general topology but i'm confused about closed discrete sets . can any one please help me out about that what type of the sets are closed discrete sets in general topology?

Answer 1

A set $C \subseteq X$ is closed and discrete when

• for all $x \in C$ we have an open set $O$ containing $x$ such that $O \cap C= \{x\}$ (discreteness)
• for all $x \notin C$ we have an open set $O$ containing $x$ such that $O \cap C = \emptyset$ (closedness).

Summarised: for all $x \in X$ we have an open neighbourhood of $x$ such that $O \cap C \subseteq \{x\}$.

Examples include all finite sets in a $T_1$ space, the integers $\mathbb{Z}$ in $\mathbb{R}$ in the usual topology, $\{\frac{1}{n}: n=1,2,3,4,\ldots\} \subseteq (0,1]$ in the usual topology.

A space is countably compact iff it has no infinite closed discrete subspace.

In a metric space $X$, $X$ is separable iff all closed and discrete subspaces are at most countable.