Path Connected

Definition: A space $$X$$ is path-connected, or PC, if for all $$p,q ∈ X$$, there is a continuous function $$f: I → X $$ such that $$f(0) = p$$ and $$f(1) = q$$ (that is, there is a path from $$p$$ to $$q$$).

A continuous image of a path-connected space is path-connected.

Path-connectedness is a homeomorphism invariant.

If $$X_α$$ is a path-connected space for all $$α$$, then the product space $$∏_α X_α$$ is path-connected. If $$X$$ is a path-connected space and $$∼$$ is an equivalence relation on $$X$$, then the quotient space $$X/∼$$ is path-connected.

Path-connectedness is not preserved by subspaces or continuous preimages.

If $$X$$ is a path-connected space, then $$X$$ is connected. Connectedness does not imply path-connectedness. In particular, the flea-and-comb space is connected but not path-connected.

A subspace $$Y$$ of $$(ℝ,𝒯_{\rm Eucl})$$ is path-connected if and only iff $$Y$$ is either an interval, ray, or $$ℝ$$.

Path-connectedness is a homotopy invariant.

If $$X$$ is a path-connected space, then $$π_1(X)$$ is independent of basepoint, up to isomorphism.

A space $$X$$ is 0-connected if $$X$$ is path-connected. A space $$X$$ is 1-connected, or simply connected, if $$X$$ is path-connected and $$π_1(X) = 1.$$

If $$X$$ and $$Y$$ are path-connected spaces and $$X ≃ Y$$, then $$π_1(X) ≅ π1(Y)$$.

If $$X$$ and $$Y$$ are homotopy equivalent path-connected spaces, then $$π_1(X)$$ is abelian [respectively, finite] if and only if $$π_1(Y)$$ is abelian [respectively, finite].

872:

If $$X$$ is a path-connected space, then $$H_0^{\rm sing}(X) ≅ ℤ$$.

A CW complex $$X$$ is path-connected if and only if the 1-skeleton $$X^{(1)}$$ is path-connected.

For any PC CW complex $$X$$, $$π_1(X) ≅ π_1(X^{(2)})$$