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## 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.

## Homework Equations

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.