indiscrete topology

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

$$\tau =\{X,\mathrm{\varnothing }\}$$

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 topologygives $X$ 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 $d(x,y)=0$.