Properties of The Fine Topology
The fine topology is in some ways much less tractable than the usual topology in euclidean space, as is evidenced by the following (taking ):
- A set in is fine compact if and only if is finite.
- The fine topology on is not locally compact (although it is Hausdorff).
- The fine topology on is not first-countable, second-countable or metrisable.
The fine topology does at least have a few 'nicer' properties:
- The fine topology has the Baire property.
- The fine topology in is locally connected.
The fine topology does not possess the Lindelöf property but it does have the slightly weaker quasi-Lindelöf property:
- An arbitrary union of fine open subsets of differs by a polar set from some countable subunion.
Read more about this topic: Fine Topology (potential Theory)
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