Topology, Discrete and Indiscrete

Discrete and Indiscrete Topologies

The discrete topology has every set open and closed. It is enough to show each point is open. The points become the base for the discrete topology. In fact, no other base will do.

Given a space T, with its own topology, a set of points S embedded in T is discrete if each point in S is contained in an open set that contains none of the other points of S. In other words, the subspace S inherits the discrete topology from T. Every point in S is its own open set. The integers are discrete in the reals, but the rationals are not.

In the indiscrete topology, only the empty set and the entire set are open and closed.

In the cofinite topology, all finite sets are closed, along with the entire set (as required). If the entire set is finite, the cofinite topology gives the discrete topology.