Pseudocircle

{{Short description|Four-point non-Hausdorff topological space}}

{{distinguish|Quasicircle}}

The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology:

$$\backslash \{\backslash \{a,b,c,d\backslash \},\; \backslash \{a,b,c\backslash \},\; \backslash \{a,b,d\backslash \},\; \backslash \{a,b\backslash \},\; \backslash \{a\backslash \},\; \backslash \{b\backslash \},\; \backslash varnothing\backslash \}.$$

This topology corresponds to the partial order X is highly pathological from the usual viewpoint of general topology, as it fails to satisfy any separation axiom besides T₀. However, from the viewpoint of algebraic topology, X has the remarkable property that it is indistinguishable from the circle S¹. More precisely, the continuous map $f$ from S¹ to X (where we think of S¹ as the unit circle in $\backslash Reals^2$) given by

is a weak homotopy equivalence; that is, $f$ induces an isomorphism on all homotopy groups. It followsAllen Hatcher (2002) [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology], Proposition 4.21, Cambridge University Press that $f$ also induces an isomorphism on singular homology and cohomology, and more generally an isomorphism on all ordinary or extraordinary homology and cohomology theories (e.g., K-theory).

This can be proven using the following observation. Like S¹, X is the union of two contractible open sets {a,b,c} and {a,b,d} whose intersection {a,b} is also the union of two disjoint contractible open sets {a} and {b}. So, like S¹, the result follows from the groupoid Seifert-van Kampen theorem, as in the book Topology and Groupoids.Ronald Brown (2006) [http://pages.bangor.ac.uk/~mas010/topgpds.html "Topology and Groupoids"], Bookforce

More generally, McCord has shown that, for any finite simplicial complex K, there is a finite topological space X_K which has the same weak homotopy type as the geometric realization |K| of K. More precisely, there is a functor taking K to X_K, from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence from |K| to X_K.{{cite journal |last=McCord |first=Michael C. |year=1966 |title=Singular homology groups and homotopy groups of finite topological spaces |journal=Duke Mathematical Journal |doi=10.1215/S0012-7094-66-03352-7 |volume=33 |issue=3 |pages=465–474}}