# Every locally path connected space has a basis consisting of path connected sets

Esran

## Homework Statement

The definition for local path connectedness is the following: let x be in X. Then for each open subset U of X such that x is in U, there exists an open V contained in U such that x is in V and the map induced by inclusion from the path components of V to the path components of U is trivial.

We wish to show that every locally path connected space has a basis of path connected sets which generates its topology.

Not applicable.

## The Attempt at a Solution

Normally, I'd be working with the more standard definition that there exists a neighborhood about each x in X containing a path connected open set, say U. Then, the desired basis would just be the collection of all such sets U for all x in X. But, I'm not sure the definition provided in class is equivalent to the more standard definition--I think there might be a mistake in it, as in I copied it down wrong. At least, I'm having extreme difficulties proving the class definition implies the more standard one.

Staff Emeritus
What happens if you take $$\mathcal{B}(x)$$ all the open neighbourhoods around x which are path connected? Try to show that this is a basis?