nLab finite topological space




topology (point-set topology, point-free topology)

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topological homotopy theory




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



Every finite topological space is an Alexandroff space, i.e. finite topological spaces are equivalent to finite preordered sets, by the specialisation order.


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 2\mathbf{2} is Sierpinski space (two points 00, 11 and three opens \emptyset, {1}\{1\}, and {0,1}\{0, 1\}), then the continuous map I=[0,1]2I = [0, 1] \to \mathbf{2} taking 00 to 00 and t>0t \gt 0 to 11 is a weak homotopy equivalence1.

The essential construction in the proof is as follows: for any finite topological space XX with specialization order 𝒪(X)\mathcal{O}(X), the topological interval map I2I \to \mathbf{2} induces a weak homotopy equivalence B𝒪(X)XB\mathcal{O}(X) \to X:

B𝒪(X)= [n]ΔCat([n],𝒪(X))Int([n],I) [n]ΔCat([n],𝒪(X))Int([n],2)XB\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 Δ op\Delta^{op} with the category IntInt of finite intervals with distinct top and bottom, so that [n]Int([n],I)[n] \mapsto Int([n], I) is a covariant functor on Δ\Delta). A few remarks on this construction:

  • The interval [n][n] has n+2n+2 elements, two of which are the distinct top and bottom. The space Int([n],I)Int([n], I) is the nn-dimensional affine simplex. The space Int([n],2)Int([n], \mathbf{2}) has n+1n+1 points 0,1,,n0, 1, \ldots, n, where jj is in the closure of j+1j+1 for 0j<n0 \leq j \lt n. The map Int([n],I)Int([n],2)Int([n], I) \to Int([n], \mathbf{2}) induced by I2I \to \mathbf{2} takes every interior point of Int([n],I)Int([n], I) to nInt([n],2)n \in Int([n], \mathbf{2}).

  • Informally, the isomorphism on the right says that any finite topological space XX can be constructed by gluing together copies of Sierpinski space 2\mathbf{2}, just as any preorder can be constructed by gluing together copies of the preorder {01}\{0 \leq 1\}. More formally, the isomorphism is established for objects XX in the equivalent category PreOrd finPreOrd_{fin}, by restricting an isomorphism over objects XX of the larger category PreOrdPreOrd, given by the counit of a nerve and realization adjunction

    [n]ΔCat([n],X)Int([n],{01}) [n]ΔCat([n],X)[n]counitX\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 PreOrdCatnerveSet Δ opPreOrd \hookrightarrow Cat \stackrel{nerve}{\hookrightarrow} Set^{\Delta^{op}} are fully faithful.

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



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 LL with a bottom element \bot, for which there exists a continuous path α:IL\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×Lα×1L×LLI \times L \stackrel{\alpha \times 1}{\to} L \times L \stackrel{\wedge}{\to} L.

Last revised on July 27, 2022 at 22:54:48. See the history of this page for a list of all contributions to it.