nLab
finite topological space

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 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.

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). 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 {01}\{0 \leq 1\}.

On the other hand, any finite simplicial complex XX 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

References

A survey 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 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.

Revised on August 24, 2016 14:04:14 by Urs Schreiber (89.204.138.57)