gn.general topology - A question about local connectedness

Any indecomposable continuum has the property you desire.

(A continuum is indecomposable if it cannot be written as a union of two proper subcontinua.)

One way to see this is that any indecomposable continuum has uncountably many composants, all of which are mutually disjoint, and all of which are dense in $x$. (Here the composant of a point $x$ in $X$ is the union of all proper subcontinua of $X$ that contain $x$.)

Here is a more direct proof: Suppose that $C$ is a proper subcontinuum of $X$ that is a neighborhood of $x$. Then every connected component of $V := Xsetminus C$ contains a point of $C$ in its boundary. (This is known as the 'boundary bumping theorem'.)

If $overline{V}$ is connected, then $overline{V}$ and $C$ are proper subcontinua of $X$ whose union is $X$.

If $V$ is disconnected, decompose $V$ into two relatively closed disjoint subsets $A$ and $B$; then $Acup C$ and $Bcup C$ are the desired subcontinua.

A simple example of an indecomposable continuum is given by the Knaster bucket-handle, see

http://commons.wikimedia.org/wiki/File%3aThe_Knaster_%22bucket-handle%22_continuum.svg .

The solenoid, mentioned in another answer, is another indecomposable continuum. You can also get such examples from "Lakes of Wada" continua. Of course the double Cantor brush given by Victor is not indecomposable (and in fact hereditarily decomposable).