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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``Re: Added definition of locally simply connected''
by antonio on 2003-02-05 15:33:39 |
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| I was tempted to define "locally" as it's used in point set topology.
Roughly, given a topological property $P$ (which a space may or may not satisfy), we say that $X$ satisfies $P$ locally at $x$ if every open neighborhood of $x$ contains a neighborhood $V$ of $x$ which satisfies $P$.
This works for many values of $P$, such as "connected", "simply connected", "path connected", "Euclidean", etc. But it doesn't work for the important example of "locally compact".
One way to fix it is to allow $V$ to be a not-necessarily-open neighborhood of $x$, that is, a subspace which contains an open neighborhood of $x$. This would then make the locally compact case work.
However, I'm afraid that it might break the definition for "locally connected," for instance, though I'm not sure. The reason I'm suspicious is that it's not true that if a subspace of $X$ is connected then so is its interior.
Anyway. Maybe it's worth adding an entry with the caveat that it's not _always_ used like that. Any thoughts welcome.
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