Contents

# Contents

## Definition

###### Definition

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

## Properties

###### Proposition

Every finite topological space is an Alexandroff space, i.e. 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 67).

###### Proof (sketch)

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

The essential construction in the proof is as follows: for any finite topological space $X$ with specialization order $\mathcal{O}(X)$, the topological interval map $I \to \mathbf{2}$ induces a weak homotopy equivalence $B\mathcal{O}(X) \to X$:

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

(where we implicitly identify $\Delta^{op}$ with the category $Int$ of finite intervals with distinct top and bottom, so that $[n] \mapsto Int([n], I)$ is a covariant functor on $\Delta$). A few remarks on this construction:

• The interval $[n]$ has $n+2$ elements, two of which are the distinct top and bottom. The space $Int([n], I)$ is the $n$-dimensional affine simplex. The space $Int([n], \mathbf{2})$ has $n+1$ points $0, 1, \ldots, n$, where $j$ is in the closure of $j+1$ for $0 \leq j \lt n$. The map $Int([n], I) \to Int([n], \mathbf{2})$ induced by $I \to \mathbf{2}$ takes every interior point of $Int([n], I)$ to $n \in Int([n], \mathbf{2})$.

• Informally, the isomorphism on the right says that any finite topological space $X$ can be constructed by gluing together copies of Sierpinski space $\mathbf{2}$, just as any preorder can be constructed by gluing together copies of the preorder $\{0 \leq 1\}$. More formally, the isomorphism is established for objects $X$ in the equivalent category $PreOrd_{fin}$, by restricting an isomorphism over objects $X$ of the larger category $PreOrd$, given by the counit of a nerve and realization adjunction

$\int^{[n] \in \Delta} Cat([n], X) \cdot Int([n], \{0 \leq 1\}) \cong \int^{[n] \in \Delta} Cat([n], X) \cdot [n] \stackrel{counit}{\cong} X$

where the counit is an isomorphism because the inclusions $PreOrd \hookrightarrow Cat \stackrel{nerve}{\hookrightarrow} Set^{\Delta^{op}}$ are fully faithful.

On the other hand, any finite simplicial complex $K$ is homotopy equivalent to its barycentric subdivision. This is $B P K$, the geometric realization of the nerve of the poset $P K$ whose elements are simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.

## Examples

A survey which includes the McCord theorems as background material is in

• Jonathan Barmak, Topología Algebraica de Espacios Topológicos Finitos y Aplicaciones PhD thesis 2009 (pdf)

published as

• Jonathan Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in Mathematics,2032. Springer, Heidelberg (2011).

The original results by McCord are in

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

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

Generalization to ringed finite spaces is discussed in

and aspects of their homotopy theory is discussed in

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