Are there spaces that are globally Euclidean, but locally not? I was reading this question, and I idly wondered if there could be spaces that were globally Euclidean, but locally not, and couldn't think of any circumstances in which that might be the case.
As the comments have suggested, the question is a bit under-determined, but let me supply some definitions in a plausible way and answer it based on those definitions.

Let $P$ be a property of topological spaces.  (Formally speaking this could be any "predicate", but it would only be asking for trouble if $P$ were not a topological property of topological spaces -- i.e., such that if $X$ and $Y$ are homeomorphic topological spaces, then $X$ has property $P$ iff $Y$ has property $P$.)

Then we say that $X$ is locally P if for every point of $x$ there is a neighborhood base $\{U_x\}$ each of whose elements $U_x$ has property $P$ in the subspace topology.  

We say that $X$ is globally P if it has property $P$.  Why would we say this?  Well, it's a matter of emphasis, presumably by way of comparison to something which locally has property $P$.

Now define a space to be Euclidean if it is homeomorphic to $\mathbb{R}^n$ for some $n \in \mathbb{N}$.

Now I claim that any space which is globally Euclidean is also locally Euclidean.  In other words, I claim that for all $n \in \mathbb{N}$, every $x \in \mathbb{R}^n$ admits a neighborhood base, each of whose members is homeomorphic to Euclidean $m$-space for some $m$.  But indeed, for any $\epsilon &gt; 0$, the open $\epsilon$-ball around $x$ is isomorphic to $\mathbb{R}^n$.

Note that in general "globally P" need not imply "locally P".  For instance a connected space need not be locally connected, and a quasi-compact (non-Hausdorff!) space need not be locally quasi-compact.  

In fairness I should say that some mathematicians don't like the idea that globally $P$ need not imply locally $P$, so they weaken the definition of locally $P$ to "each point of $X$ admits some neighborhood which has property $P$".  I would call this weakly locally P and then it is trivial that globally $P$ implies weakly globally $P$: for 
each $x$, take $X$ itself as a neighborhood which has property $P$.
Just another attempt could be:
Define sth is locally adjective if it admits a neighborhood base consisting of adjective neighborhoods at every point and sth is globally adjective if it admits a basis for the topology consisting of adjective opens...

However, that would be again just seeking for some consistency pressing in some technical dress, so hmm I think best is still to stick with the luckily almost widely accepted aggreement presented by Pete Clark - though I like Andrew's view on these things although it's kind of vague.