Trivial topology

The trivial topology, indiscrete topology or lumps topology is considered in the mathematical branch of topology structure for a lot that makes this a topological space.

Definition

Be a lot. The trivial topology is the topology in which only the amount and the empty set are open.

Properties

Be a topological space equipped with the trivial topology.

• All points in are topologically indistinguishable.
• According to the definition, only the empty set and the whole lot have been completed.
• The amount is the only basis of.
• The space is compact and thus in particular paracompact, Lindelöf and locally compact.
• The space is path-connected, for each image of a topological space after is continuous, and therefore contiguous.
• The trivial topology is the coarsest of all topologies on a given set, in particular any map from a topological space into a trivial topology is continuous.
• The trivial topology has all the usual separation properties, provided they do not require T ₀, and associating the pseudo pseudometrisierbar metric that any two points the distance between 0.
• Each filter converges to the trivial topology against any point, this characterizes the trivial topology.