Locally Connected Space - Quasicomponents

Quasicomponents

Let X be a topological space. We define a third relation on X: if there is no separation of X into open sets A and B such that x is an element of A and y is an element of B. This is an equivalence relation on X and the equivalence class containing x is called the quasicomponent of x.

can also be characterized as the intersection of all clopen subsets of X which contain x. Accordingly is closed; in general it need not be open.

Evidently for all x in X. Overall we have the following containments among path components, components and quasicomponents at x:

If X is locally connected, then, as above, is a clopen set containing x, so and thus . Since local path connectedness implies local connectedness, it follows that at all points x of a locally path connected space we have

Examples

1. An example of a space whose quasicomponents are not equal to its components is a countable set, X, with the discrete topology along with two points a and b such that any neighbourhood of a either contains b or all but finitely many points of X, and any neighbourhood of b either contains a or all but finitely many points of X. The point a lies in the same quasicomponent of b but not in the same component as b.

2. The Arens–Fort space is not locally connected, but nevertheless the components and the quasicomponents coincide: indeed for all points x.