Local Homeomorphisms: Characterization

Question

Problem

Consider for simplicity a surjection $F:X\to Y$.

Are these characterizations of local homeomorphisms equivalent: $$\forall x\in X:\exists U_x\in\mathcal{T}_X, V_y\in\mathcal{T}_Y:\quad U_x\cong V_y\quad (F(x)=y)$$ $$\forall y\in Y:\exists U_x\in\mathcal{T}_X, V_y\in\mathcal{T}_Y:\quad U_x\cong V_y\quad (F(x)=y)$$ (I'm asking as usually this is handled rather sloppy.)

Disclaimer

This thread is meant as summary. For more informations see:

• SE blog: Answer own Question
• MSE meta: Answer own Question

(The second especially reveals the opinion of the community!)

Answer 1

Inclusion

The former implies the latter as: $$y=F(x)\implies U_x\cong V_y$$ (That is one can choose any preimage.)

Counterexample

Consider the wrap of the real line onto the dotted circle: $$F:\mathbb{R}\to\mathbb{S}\setminus\{1\}:\quad F(\varphi=0):=-1,F(\varphi\neq0):=e^{i\varphi}$$ Then only the ladder holds true since: $$F(\pi-\varepsilon,\pi+\varepsilon)\cong(\pi-\varepsilon,\pi+\varepsilon)\quad(\varepsilon\leq\pi)$$ $$F(0-\varepsilon,0+\varepsilon)\ncong(0-\varepsilon,0+\varepsilon)\quad(\varepsilon\leq\pi)$$ (That is one may have to choose specific preimage.)

Perspective

Note that a covering handles precisely this discrepancy!