Trivial topology

The trivial topology , indiscreet topology , chaotic topology or cluster topology is a structure considered in the mathematical sub-area of topology for a set that makes it a topological space .

definition

Be a lot. The trivial topology on is the topology where only the set and the empty set are open . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \emptyset }$

properties

Let a topological space be provided with the trivial topology. ${\displaystyle X}$

• All points in are topologically indistinguishable . ${\displaystyle X}$
• According to the definition, only the empty set and the whole set are closed . ${\displaystyle X}$
• The space is compact and therefore paracompact , lindelöf and locally compact . ${\displaystyle X}$
• The space is path-connected , because every mapping of a topological space is continuous and therefore also connected . ${\displaystyle X}$ ${\displaystyle X}$
• The trivial topology is the coarsest of all topologies on a given set; in particular, every mapping from a topological space into a trivial topology is continuous .
• The trivial topology has all the usual separation properties , unless they require T₀ , and can be pseudometrized by the pseudometrics, which assigns the distance 0 to any two points.
• Each filter converges towards each point in the trivial topology; this characterizes the trivial topology.

See also

• Discrete topology

Individual evidence

1. ↑ Trivial topology. In: Guido Walz (Red.): Lexicon of Mathematics. Volume 5: Sed to Zyl. Spectrum - Akademischer Verlag, Heidelberg et al. 2002, ISBN 3-8274-0437-1 .
2. ↑ Lothar Tschampel: BUCHMAT. 3.A: Topology 1. General topology. Version 2. Buch-X-Verlag, Berlin 2011, ISBN 978-3-934671-60-7 .
3. ↑ Harro Heuser : Textbook of Analysis . 13th revised edition. tape 2 . Teubner, Stuttgart et al. 2004, ISBN 3-519-62232-7 , pp. 210 .
4. ^ ^{A b} Gerd Laures, Markus Szymik: Basic course topology . Spectrum - Akademischer Verlag, Heidelberg 2009, ISBN 978-3-8274-2040-4 , p. 9 .