Context: I am given, as an exercise, the task of proving this claim:
For $X$ a topological space, show it is path-connected if and only if $X$ is connected and each $x \in X$ lies in some path-connected open neighborhood (presumably of $x$).
Proving the forward implication is straightforward enough, but I've been having difficulties with the converse.
First, some definitions for context:
$ \newcommand{\nc}{\newcommand} \nc{\T}{\mathcal{T}} \nc{\p}[1]{\left(#1_1,#1_2\right)} $ Definitions: We assume throughout $(X,\mathcal{T})$ is a topological space.
$X$ is path-connected if $\forall x,y \in X$ there exists $\gamma : [0,1] \to X$ continuous such that $\gamma(0) = x$ and $\gamma(1) = y$.
$X$ is connected if it is not disconnected.
- That is, $X$ is connected if we cannot write it as a disjoint union of nonempty open sets.
- That is, symbolically, $X$ is connected if we cannot write $X = A \cup B$ for $\varnothing \ne A,B \in \mathcal T$ with $A \cap B = \varnothing$.
$X$ is locally connected if $\mathcal{T}$ has a basis of connected sets.
Note that if $X$ is path-connected, it is also connected.
Elaboration: As I looked into the question, I came across a case (thanks to this post) that would contradict the possible alternative claim where $$ \text{locally connected & connected} \implies \text{path-connected} $$ This counterexample would arise with the lexicographic ordering on $[0,1]^2$.
For $x := \p \xi,y := \p \eta \in [0,1]^2$ we say $$ x \prec y \iff (\xi_1 < \eta_1) \lor (\xi_1 = \eta_1, \xi_2 < \eta_2) $$ This induces a topology $\T$ on $[0,1]^2$ with the basis formed by open "intervals"
$$(x,y) := \left\{ z := \p \zeta \in [0,1]^2 \, \middle| \, x \prec z \prec y \right\}$$
To my understanding, this topology $\T$ is locally-connected and connected, yet not path connected -- at least from what I can look up.
Why is this relevant? Well, my thinking is this: if each $x \in X$ is in some path-connected open neighborhood, I feel like that might imply that $X$ is locally connected. That's where my instincts lie. Then the above example would give a counterexample.
My Question: Does this prove to be a suitable counterexample for the converse of the proposed claim I'm told to prove?
More directly, then: is it true to say that, if for each $x \in X$ (for $(X,\T)$ any connected topological space), there exists $C_x \in \T$ path-connected containing $x$, then we have local connectedness?
If not (and there is no suitable counterexample), what might be a suitable way to indeed prove the converse of the claim? I've been having trouble doing so -- in particular, I have no clue how I should establish an $x,y$-path function $\gamma$ in $X$ if the claim is true (how would one deal with the case when $x,y$ are in different such neighborhoods in the hypothesis?).
