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.
Best Answer
In the finite complement topology, and infinite subset of the same size is homeomorphic to the whole space (any bijection between finite complement topologies is a homeomorphism).
And injective mapping from $[0,1]$ (usual topology) into a space with the finite complement topology is continuous.
The second fact implies the path-connectedness, the first implies the local path-connectedness. QED.