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.

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.

 

[micromass]

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?

What definition of basis are you using by the way?

 

[Esran]

The difficulty though is first proving that with the given definition of local path connectedness, x has such a neighborhood. After that, I know exactly what to do. Going from the class definition to the more standard one is my problem.

 

[Esran]

My definition of basis is as follows:

The base elements cover X. Let B1, B2 be base elements and let I be their intersection. Then for each x in I, there is a base element B3 containing x and contained in I.