---
tag: MGE14ST1YP
kind: concept
name: Alexander–Spanier cohomology
msc: [55N05, 55N30]
sources:
  - url: https://en.wikipedia.org/wiki/Alexander–Spanier_cohomology
same_as:
  wikidata: Q4143007
  wikipedia: Alexander–Spanier cohomology
---

## Summary

Alexander–Spanier cohomology is a cohomology theory for topological spaces built from arbitrary functions on tuples of points, taken modulo functions that vanish near the diagonal.
It does not use simplices or maps from a model space, so it behaves well on spaces with bad local structure.
On paracompact Hausdorff spaces it agrees with Čech cohomology and with sheaf cohomology of a constant sheaf.
On manifolds and CW complexes it agrees with [[singular cohomology|6N7BC70CX0]].

## Explanations

### L1 human

Singular cohomology probes a space with maps of triangles into it.
On wild spaces such maps may miss features that you can see by eye.
Alexander–Spanier cohomology instead looks at functions of several points that only matter when the points are close together.
For example, the Warsaw circle is a circle with a wild oscillating piece.
It has no first singular cohomology, but its Alexander–Spanier cohomology sees the loop, as it does for an ordinary circle.