The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology: \{\{a,b,c,d\}, \{a,b,c\}, \{a,b,d\}, \{a,b\}, \{a\}, \{b\}, \varnothing\}.
The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology: \{\{a,b,c,d\}, \{a,b,c\}, \{a,b,d\}, \{a,b\}, \{a\}, \{b\}, \varnothing\}.
This topology corresponds to the partial order a where the open sets are downward-closed sets. 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 \Reals^2) given by f(x,y) = \begin{cases}a,& x0\\ c,& (x,y)=(0,1)\\ d,& (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 ordinary or extraordinary homology and cohomology theories (e.g., K-theory).
Discovered by embedding cosine similarity (sentence-transformers MiniLM, 384-dim).