CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The pseudocircle is a finite topological space which is weakly homotopy equivalent to the standard circle.
In other word, for the purposes of homotopy theory (up to weak homotopy equivalence), it is equivalent to the circle, yet it is a purely finite combinatorial creature.
The pseudocircle $\mathbb{S}$ is the topological space
whose underlying set is a $4$-set, say $\{l,r,t,b\}$ (for the โleft sideโ, โright sideโ, โtop pointโ, and โbottom pointโ of the circle)
and whose topological structure, given as the collection of open subsets, is
That is the topology generated by the base $\{\{l,r,t\}, \{l,r,b\}, \{l\}, \{r\}\}$ (which is in fact the unique minimal base and furthermore the unique minimal subbase).
As a frame, this topology is a subframe of the frame of opens of the usual circle, where the names of $l$, $r$, $t$, and $b$ are taken literally.
These also name a partition of the standard circle, and this gives a quotient map? from the circle to the pseudocircle; this map is the promised weak homotopy equivalence, see below.
The function of sets
from the standard circle to the pseudocircle, which sends
a point in the open left half of $S^1$ (thought of under the standard embedding into $\mathbb{R}^2$) to the point $l \in \mathbb{S}$;
a point in the open right half of $S^1$ to $r \in \mathbb{S}$;
the top point of $S^1$ to $t \in \mathbb{S}$;
the bottom point of $S^1$ to $b \in \mathbb{S}$
is a continuous function. Moreover, it is a weak homotopy equivalence.
See the proof of the general statement at finite topological space - properties.
The only continuous functions in the other direction, $\mathbb{S} \to S^1$ are the constant maps.