---
tag: W2ZD4XP74R
kind: concept
name: Fine topology
msc: [31C40, 31B05]
sources:
  - url: https://en.wikipedia.org/wiki/Fine_topology_(potential_theory)
same_as:
  wikidata: Q5450128
  wikipedia: Fine topology (potential theory)
---

## Summary

The fine topology on $\R^n$ is the coarsest topology that makes every [[superharmonic function|X4V2S56YQQ]] continuous.
It is strictly finer than the Euclidean topology.
Henri Cartan introduced it to study [[thin sets|7DGF6MMEGX]]: a set is a fine neighbourhood of a point exactly when its complement is thin there.
Many results of potential theory, such as the behaviour of solutions at irregular boundary points, are stated most cleanly with fine limits.

## Explanations

### L1 human

Superharmonic functions can be discontinuous in the usual sense, for instance a Newtonian potential that is infinite on a sequence of points.
The fine topology adds just enough open sets to make all of them continuous.
In this topology, removing a set of capacity zero, such as a sequence of points, from an open set leaves a set that is still open.
So the fine topology ignores sets that harmonic functions cannot see.