Page "Stone duality" Paragraph 14

Let us first look at the points of a topological space X.

One is usually tempted to consider a point of X as an element x of the set X, but there is in fact a more useful description for our current investigation.

Any point x gives rise to a continuous function p < sub > x </ sub > from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p < sub > x </ sub >( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.

Therefore the set of points of a topological space is equivalently characterized as the set of functions from 1 to X.