Download PDF

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Geometrization conjecture wikipedia, lookup

Brouwer fixed-point theorem wikipedia, lookup

Continuous function wikipedia, lookup

Sheaf (mathematics) wikipedia, lookup

Surface (topology) wikipedia, lookup

Covering space wikipedia, lookup

3-manifold wikipedia, lookup

Fundamental group wikipedia, lookup

Grothendieck topology wikipedia, lookup

General topology wikipedia, lookup

Transcript
clopen subset∗
mathcam†
2013-03-21 15:35:58
A subset of a topological space X is called clopen if it is both open and
closed.
Theorem 1. The clopen subsets form a Boolean algebra under the operation of
union, intersection and complement. In other words:
• X and ∅ are clopen,
• the complement of a clopen set is clopen,
• finite unions and intersections of clopen sets are clopen.
Proof. The first follows by the definition of a topology, the second by noting
that complements of open sets are closed, and vice versa, and the third by noting
that this property holds for both open and closed sets.
One application of clopen sets is that they can be used to describe connectness. In particular, a topological space is connected if and only if its only clopen
subsets are itself and the empty set.
If a space has finitely many connected components then each connected
component is clopen. This may not be the case if there are infinitely many
components, as the case of the rational numbers demonstrates.
∗ hClopenSubseti
created: h2013-03-21i by: hmathcami version: h33978i Privacy setting:
h1i hDefinitioni h54D05i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1