Sierpiński space

Example

Let $S = \set{0, 1}$ with the family of open sets $\set{\emptyset, \set{1}, \set{0, 1}}$. This family contains $\emptyset$ and $S$, and it is closed under unions and intersections, as one checks on its three members. So $S$ is a topological space, the Sierpiński space.

• $S$ is a T0 space: the open set $\set{1}$ contains $1$ but not $0$.
• $S$ is not a T1 space: the only open set that contains $0$ is $S$, and it also contains $1$.
• The closed sets are $\emptyset$, $\set{0}$ and $S$, so the closure of $\set{1}$ is $S$.
• $S$ is connected, since its only clopen sets are $\emptyset$ and $S$.

For any topological space $X$, a map $f \colon X \to S$ is continuous if and only if $f^{-1}(\set{1})$ is open in $X$. So continuous maps $X \to S$ correspond one to one to open subsets of $X$.