CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The Sierpiński space $\Sigma$ is the topological space which is the set of truth values, classically $\{\bot, \top\}$, equipped with the specialization topology, in which $\{\bot\}$ is closed and $\{\top\}$ is open but not conversely. (The opposite convention is also used.)
In constructive mathematics, it is important that $\{\top\}$ be open (and $\{\bot\}$ closed), rather than the other way around. Indeed, the general definition (since we can't assume that every element is either $\top$ or $\bot$) is that a subset $P$ of $\Sigma$ is open as long as it is upward closed: $p \Rightarrow q$ and $p \in P$ imply that $q \in P$. The ability to place a topology on $\Top(X,\Sigma)$ is fundamental to abstract Stone duality, a constructive approach to general topology.
This Sierpinski space
is a contractible space;
has a focal point.
According properties are inherited by the Sierpinski topos and the Sierpinski (∞,1)-topos over $Sierp$.
The Sierpinski space is a classifier for closed subspaces of a topological space $X$ in that for any closed subspace $A$ of $X$ there is a unique continuous function $\chi_A: X \to S$ such that $A = \chi_A^{-1}(\bot)$.
Dually, it classifies open subsets in that any open subspace $A$ is $\chi_A^{-1}(\top)$. Note that the closed subsets and open subsets of $X$ are related by a bijection through complementation; one gets a topology on the set of either by identifying the set with $\Top(X,\Sigma)$ for a suitable function space topology.