Non- intuitive connected space.

Question

There exist knowing examples of connected spaces such that its picture is a counter intuitive for us?. I mean a topology on a set who makes see the space as connected (no connected) but it is no connected (connected)?

Thanks!

Answer 1

As suggested by Mirko Swirko, you may want to look at indecomposable continua. A continuum is a compact connected Hausdorff space. A continuum is indecomposable if it is not the union of two proper subcontinua.

The following theorem says that we can "almost" disconnect an indecomposable continuum. Specifically, we can write $X$ as the union of two closed sets whose intersection is contained in an given open set.

Theorem. If $X$ is an indecomposable continuum then for each open set $U$ there exists two nonempty closed sets $A$ and $B$ with $X=A\cup B$ and $A\cap B\subseteq U$.

Another theorem. If $X$ is a metric indecomposable continuum then we can partition $X$ into $2^\omega $ many dense subsets such that any proper closed subset of $X$ intersecting two of these partition sets is not connected!

The simplest example of a metric indecomposable continuum is the Knaster "buckethandle" continuum (google it).