Locally finite collection
A collection of subsets of a topological space is said to be locally finite if each point in the space has a
In the
Note that the term locally finite has different meanings in other mathematical fields.
Examples and properties
A finite collection of subsets of a topological space is locally finite.[2] Infinite collections can also be locally finite: for example, the collection of subsets of of the form for an integer .
Every locally finite collection of sets is
If a collection of sets is locally finite, the collection of the
An arbitrary union of closed sets is not closed in general. However, the union of a locally finite collection of closed sets is closed.[4] To see this we note that if is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood of that intersects this collection at only finitely many of these sets. Define a
In compact spaces
Every locally finite collection of sets in a compact space is finite. Indeed, let be a locally finite family of subsets of a compact space . For each point , choose an
In Lindelöf spaces
Every locally finite collection of sets in a Lindelöf space, in particular in a second-countable space, is countable.[5] This is proved by a similar argument as in the result above for compact spaces.
Countably locally finite collections
A collection of subsets of a topological space is called σ-locally finite[6][7] or countably locally finite[8] if it is a countable union of locally finite collections.
The σ-locally finite notion is a key ingredient in the
In a Lindelöf space, in particular in a second-countable space, every σ-locally finite collection of sets is countable.
Citations
- ^ a b Munkres 2000, p. 244.
- ^ Munkres 2000, p. 245 Lemma 39.1.
- ^ Engelking 1989, Theorem 1.1.13.
- ^ Engelking 1989, Corollary 1.1.12.
- ^ Engelking 1989, Lemma 5.1.24.
- ^ Willard 2004, Definition 20.2.
- ^ Engelking 1989, p. 280.
- ^ Munkres 2000, p. 245.
- ^ Engelking 1989, Theorem 4.4.7.
- ^ Munkres 2000, p. 250 Theorem 40.3.
References
- ISBN 3-88538-006-4.
- Munkres, James R. (2000), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2
- Willard, Stephen (2004) [1970]. General Topology. OCLC 115240.