limit points and closure for connected sets

The below theorem shows that adding limit pointsPlanetmathPlanetmath to a connected set preserves connectedness.

Theorem 1.

Suppose A is a connected set in a topological spaceMathworldPlanetmath. If ABA¯, then B is connected. In particular, A¯ is connected.

Thus, one way to prove that a space X is connected is to find a dense subspace in X which is connected.

Two touching closed ballsPlanetmathPlanetmath in 2 shows that this theorem does not hold for the interior. Along the same lines, taking the closureMathworldPlanetmathPlanetmath does not preserve separatedness.


Let X be the ambient topological space. By assumptionPlanetmathPlanetmath, if U,VA are open and UV=A, then UV. To prove that B is connected, let U,V be open sets in B such that UV=B and for a contradition, suppose that UV=. Then there are open sets R,SX such that


It follows that (RS)B=B and (RS)B=. Next, let U~,V~ be open sets in A defined as




and as (RS)A=U~V~, it follows that U~V~=(RS)A. Then, by the properties of the closure operator,


Title limit points and closure for connected sets
Canonical name LimitPointsAndClosureForConnectedSets
Date of creation 2013-03-22 15:17:56
Last modified on 2013-03-22 15:17:56
Owner matte (1858)
Last modified by matte (1858)
Numerical id 7
Author matte (1858)
Entry type Theorem
Classification msc 54D05