# 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.