Connected space

The topologist's sine curve is connected, but it is not locally connected

As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006). Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.

However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see ). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.

There are stronger forms of connectedness for topological spaces, for instance: