# indiscrete topology

If $X$ is a set and it is endowed with a topology defined by

 $\tau=\{X,\emptyset\}$

then $X$ is said to have the indiscrete topology.

Furthermore $\tau$ is the coarsest topology a set can possess, since $\tau$ would be a subset of any other possible topology. This topology gives $X$ many properties:

• Every subset of $X$ is sequentially compact.

• Every function to a space with the indiscrete topology is continuous.

• $X$ is path connected and hence connected but is arc connected only if $X$ is uncountable or if $X$ has at most a single point. However, $X$ is both hyperconnected and ultraconnected.

• If $X$ has more than one point, it is not metrizable because it is not Hausdorff. However it is pseudometrizable with the metric $d(x,y)=0$.

Title indiscrete topology IndiscreteTopology 2013-03-22 12:48:11 2013-03-22 12:48:11 mathwizard (128) mathwizard (128) 20 mathwizard (128) Definition msc 54-00 trivial topology coarse topology