Pseudocircle

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

$\left\{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\emptyset\right\}$

X is highly pathological from the 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 unit circle $S 1$ .

More precisely the map $f:S^1\longrightarrow X$ given by

Failed to parse (Cannot write to or create math output directory): f(x,y)=\begin{cases}a\quad x<0\\b\quad x>0\\c\quad(x,y)=(0,1)\\d\quad(x,y)=(0,-1)\end{cases}

is a weak homotopy equivalence, that is f induces an isomorphism on all homotopy groups. It follows that f also induces an isomorphism on singular homology and cohomology and more generally an isomorphism on all extraordinary homology and cohomology theories (e.g. K-theory).

This can be proved using the following observation. Like $S 1$ , 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 contractible open sets ${a}$ and ${b}$ .

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 $K\mapsto X_K$ from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence $|K|\longrightarrow X_K$ .