# Finite Complement Topology and Local Path Connectedness

I'm having trouble with deciding whether or not a given space is locally path connected.

Let $$(R,F)$$ denote the finite complement topological space over the real numbers.

a) Determine the connected and path-connected components of (R,F).

b) Is (R,F) locally path-connected? Prove your statements.

I have proven that the real line is connected and path-connected and so is its own component. I have also shown that it is locally connected. But my intuition fails when it comes to deciding whether or not the space is locally path-connected.

Obviously, every two points $$x, y \in R$$ can be joined by the path

$$\gamma: [0,1] \to [x,y], \gamma(t)=x+t(y-x)$$.

But if the claim were true one could show that for every $$x \in R$$ there exists a path-connected neighborhood. Since this neighborhood contains a set O which is open with respect to the Finite Complement Topology and contains x one would have to prove that O is path-connected. Since O is the real line with a finite number of points removed, I cannot imagine how such a neighborhood can exist, as my intuition fails at imagining how to continously connect the set O.

And injective mapping from $$[0,1]$$ (usual topology) into a space with the finite complement topology is continuous.