Connected space

Nov 2009
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0
Give an example of connected space but is not locally compact ?

Is it true the rational number space ?
 

Opalg

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Aug 2007
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Give an example of connected space but is not locally compact ?

Is it true the rational number space ?
The space of rational numbers is not connected. For an example of a space that is connected but not locally compact, you need to look at a space that is infinite-dimensional. Any infinite-dimensional Banach space or Hilbert space will do.
 

Drexel28

MHF Hall of Honor
Nov 2009
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Berkeley, California
Give an example of connected space but is not locally compact ?

Is it true the rational number space ?
This may be too complicated, and I can't think of any right off hand but I'm sure there it shouldn't be too hard to make one...

Take any connected space \(\displaystyle X\) which is Hausdorff but not \(\displaystyle T_\pi\) (i.e. Tychonoff space). If \(\displaystyle X\) were locally compact then it would be embeddable in \(\displaystyle X_\infty\) (i.e. the Alexandroff Compactification) and thus \(\displaystyle T_\pi\). Any normed vector space is path connected (and thus connected) but if you have an infinite dimensional Banach space it won't be

That said, any example like Opalg said would work. To see this note that in infinite dimensional Banach spaces the closure of the unit ball \(\displaystyle B_1(0)\) is not compact. Use this to show that any neighborhood of \(\displaystyle 0\) is not precompact.