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

The original results by McCord are in

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 $I \times L \stackrel{\alpha \times 1}{\to} L \times L \stackrel{\wedge}{\to} L$.