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For any property p of topological spaces, p implies locally p.

topospaces.subwiki.org. Locally operator

Locally path-connected space … This property is obtained by applying the locally operator to the property: path-connected space

topospaces.subwiki.org. Locally path-connected space

This space is obviously path-connected, but it is not locally path-connected

math.stackexchange.com

This seems like a contradiction.

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What is the question? –  Keenan Kidwell Aug 4 '12 at 18:11
    
@Keenan Kidwell: Is there a contradiction, or I can't see something? If so, where does the contradiction lie (on what page)? –  beroal Aug 5 '12 at 14:06

3 Answers 3

Locally path-connected means that for every $x\in X$ and every neighbourhood $V$ of $x$, you can find an path-connected neighbourhood $U$ of $x$ such that $U\subset V$ (and not only "for each $x\in X$ there exists an path-connected neighbourhood of $x$").

Defining in this form, it's possible to have an path-connected space which isn't locally path-connected.

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2  
This is called a strong local property in the link the OP gave. Interestingly, I was taught (and I think it is standard terminology) that what you gave is local path-connectedness and strong local path-connectedness means we can find an open path-connected neighbourhood below any neighbourhood. Confusingly, some authors (mostly in the European tradition) use local compactness in the sense of the OP. –  Miha Habič Aug 4 '12 at 18:33
    
@MihaHabič: Thanks! =p I sincerely never have seen (or never noticed) any author giving both definitions, only the strong one (I've seen too few books... orz... and I used to think: "the weak one seems useless..."). I should pay more attention from now... =p –  Yuki Aug 4 '12 at 19:24
    
Do I understand it correctly that the page with “This property is obtained by applying the locally operator” is mistaken? –  beroal Aug 5 '12 at 14:04
    
@beroal: No, it's just convention. "Locally path-connected" is usually used to mean "strong locally path-conneced" (in the terminology of the page); but in that page, they defined both notions. –  Yuki Aug 6 '12 at 2:09
    
“but in that page, they defined both notions” Both? I can't find both. Do you talk about the page with the name “Locally path-connected space”? –  beroal Aug 7 '12 at 7:59

The definition in the first link: property $p$ holds locally if for each $x \in X$ there exists a neighborhood $U \ni x$ such that $p$ holds on $U$. Then it is obvious that if $p$ holds (on the full space) it will also hold locally, just take $U = X$. An example of such a property is compactness.

Now the definition of local (path-)connectedness uses a different (stronger) notion of "locally", as Yuki spelled out. In the first link this is called a "strongly locally operator".

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In this page they explain the difference. For an example of path-connected space which is not strongly locally path-connected, consider the subspace of $[0,1] \times [0,1]$ of points whose $y$ coordinate is rational with the subspace topology. It is path-connected (use a path going to $x$ coordinate $0$ or $1$ to connect two points), but a small enough neighborhood of a point with $x$-coordinate not equal to $0$ or $1$ will not have a path-connected neighborhood because intersecting an open ball (in $[0,1] \times [0,1]$) with this subspace will give you a bunch of small line segments very close to each other, but not path-connected.

Hope that helps,

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