Finding Path Components in Topological Space X

[pivoxa15]

Homework Statement

Consider a topological space X

Show that every point of X is contained in a unique path component, which can be defined as the largest path connected subset of X containing this point.

The Attempt at a Solution

What happens if we take X=Q? There are no path connected subsets of Q. Or would in this case the path components are the sets containing the individual elements of Q? Which form the basis for Q. So since there exists a basis for every topological space, the elements of the basis of X are always path components.

 

[matt grime]

Q is not a topological space until you specify the topology.

The subset {r} is a path connected component for any rational r. They certainly do not form a basis of a topology on Q unless you're talking about the discrete topology (if you're specifying the open sets, or cofinite if you're specifying the closed sets).

There is nothing that implies a given generating set for a topology must consist of path connected components.