nLab
finite topological space

Definition

Definition

A finite topological space is a topological space whose underlying set is a finite set.

Properties

Finite topological spaces are equivalent to finite preordered sets, by the specialisation order.

Theorem

Finite topological spaces have the same weak homotopy types as finite simplicial complexes / finite CW-complexes.

This is due to McCord.

Proof (sketch)

If $2$ is Sierpinski space (two points $0$, $1$ and three opens $\varnothing$, $\{1\}$, and $\{0,1\}$), then the continuous map $I=[0,1]\to 2$ taking $0$ to $0$ and $t>0$ to $1$ is a weak homotopy equivalence¹.

For any finite topological space $X$ with specialization order $𝒪(X)$, the topological interval map $I\to 2$ induces a weak homotopy equivalence $B𝒪(X)\to X$:

$$B𝒪(X)={\int }^{[n]\in \Delta }\mathrm{Cat}([n],𝒪(X))\cdot \mathrm{Int}([n],I)\to {\int }^{[n]\in \Delta }\mathrm{Cat}([n],𝒪(X))\cdot \mathrm{Int}([n],2)\cong X$$B\mathcal{O}(X) = \int^{[n] \in \Delta} Cat([n], \mathcal{O}(X)) \cdot Int([n], I) \to \int^{[n] \in \Delta} Cat([n], \mathcal{O}(X)) \cdot Int([n], \mathbf{2}) \cong X

(where we implicitly identify ${\Delta }^{\mathrm{op}}$ with the category $\mathrm{Int}$ of finite intervals with distinct top and bottom). The isomorphism on the right says that any finite topological space can be constructed by gluing together copies of Sierpinski space, in exactly the same way that any preorder can be constructed by gluing together copies of the preorder $\{0\le 1\}$.

On the other hand, any finite simplicial complex $X$ is homotopy equivalent to its barycentric subdivision, which is the geometric realization of the poset of simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.

Examples

• pseudocircle

References

A survey is in

• Jonathan A. Barmak, Topología Algebraica de Espacios Topológicos Finitos y Aplicaciones (pdf)

The original results by McCord are in

• M.C. McCord. Homotopy type comparison of a space with complexes associated with its open covers . Proc. Amer. Math. Soc. 18 (1967), 705-708.

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

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1. Any topological meet-semilattice $L$ with a bottom element $\perp$, for which there exists a continuous path $\alpha :I\to L$ connecting $\perp$ to the top element $\top$, is in fact contractible. The contracting homotopy is given by the composite $I\times L\stackrel{\alpha \times 1}{\to }L\times L\stackrel{\wedge }{\to }L$. ↩