Particular point topology

Example

Let $X$ be a set and let $p \in X$. The particular point topology on $X$ declares a set $U \subseteq X$ open if $U = \emptyset$ or $p \in U$. This is a topology: $\emptyset$ and $X$ are open. A union of sets that contain $p$ contains $p$, and so does a finite intersection of them. So $X$ is a topological space.

• $X$ is a T0 space: let $x \ne y$, and name them so that $x \ne p$. Then $\set{p, y}$ is open, contains $y$, and does not contain $x$.
• If $X$ has at least two points, $X$ is not a T1 space: every open set that contains a point $x \ne p$ also contains $p$.
• $X$ is connected: any two nonempty open sets meet at $p$, so $X$ is not a union of two disjoint nonempty open sets.
• The set $\set{p}$ is dense, since every nonempty open set contains $p$.
• $X$ is compact if and only if $X$ is finite. Indeed, the open sets $\set{p, x}$ for $x \in X$ cover $X$, and a finite subcover exists only when $X$ is finite.