indiscrete

Indiscrete topology, also called the trivial topology, on a set X is the topology in which the only open subsets are the empty set and X itself. It is the coarsest possible topology on X, meaning no larger collection of open sets is allowed beyond ∅ and X.

Because only two sets are open, the indiscrete space is not Hausdorff or T1 when X has more than one point; in fact it is not T0 for any |X| > 1. Despite this, the space is connected and compact: any open cover must contain X, providing a trivial finite subcover. The space is path-connected as well: for any two points a and b in X, one can define a continuous map from the unit interval to X with f(0) = a and f(1) = b, since every map into an indiscrete space is continuous.

Mappings involving indiscrete spaces have simple behavior. A map from any space Y into an indiscrete space X is always continuous, because the preimage of ∅ is ∅ and the preimage of X is Y, both open in Y. A map from X to another space Y need not be continuous unless it is constant or the target space’s topology allows the required preimages to be open in X.

Products of indiscrete spaces are again indiscrete: the product topology on the product of such spaces yields only the empty set and the whole product as open. This makes the indiscrete topology a useful, if extreme, example in topology and category theory.

See also: trivial topology; discrete topology; connectedness; path-connectedness.