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 topology gives $X$ many properties:

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Every subset of $X$ is sequentially compact.

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Every function to a space with the indiscrete topology is continuous^{}.

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$X$ 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.

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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 

Canonical name  IndiscreteTopology 
Date of creation  20130322 12:48:11 
Last modified on  20130322 12:48:11 
Owner  mathwizard (128) 
Last modified by  mathwizard (128) 
Numerical id  20 
Author  mathwizard (128) 
Entry type  Definition 
Classification  msc 5400 
Synonym  trivial topology 
Synonym  coarse topology 