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.