nLab
pseudocircle

Idea

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.

Definition

Definition

The pseudocircle $𝕊$ is the topological space

• whose underlying set is a $4$-set, say $\{l,r,t,b\}$ (for the ‘left side’, ‘right side’, ‘top point’, and ‘bottom point’ of the circle)

• and whose topological structure, given as the collection of open subsets, is

  $$\{\{l,r,t,b\},\{l,r,t\},\{l,r,b\},\{l,r\},\{l\},\{r\},\varnothing \}.$$\{\{l,r,t,b\}, \{l,r,t\}, \{l,r,b\}, \{l,r\}, \{l\}, \{r\}, \empty\} \,.

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).

Remark

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.

Properties

Relation to the standard circle

Proposition

The function of sets

$$f:S^1\to 𝕊$$f : S^1 \to \mathbb{S}

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 ${ℝ}^2$) to the point $l\in 𝕊$;

• a point in the open right half of $S^1$ to $r\in 𝕊$;

• the top point of $S^1$ to $t\in 𝕊$;

• the bottom point of $S^1$ to $b\in 𝕊$

is a continuous function. Moreover, it is a weak homotopy equivalence.

See the proof of the general statement at finite topological space - properties.

Proposition

The only continuous functions in the other direction, $𝕊\to S^1$ are the constant maps.

References

• Michael McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. Volume 33, Number 3 (1966), 465-474. (EUCLID)

• Wikipedia, Pseudocircle