This is should be a simple one. I know what I'm looking for but I don't know what to call it.

I have a member on PHF that I am talking with and I'd like to know the terminology used in the next two definitions.

1) I want to compactify the real numbers by defining $\displaystyle - \infty$ and $\displaystyle \infty$ to belong to the compactified set such that, for any member "a" in the compactified set we have that $\displaystyle -\infty \leq a \leq \infty$. My model is the "hyperintegers" $\displaystyle [ -\infty, \text{ ... } -1, 0, 1, \text{ ... } , \infty ] $. I know that the hyperreals also contain infinitesimals but I don't actually need them for the discussion.

2) Can I say that the reals and the compactified real numbers defined above are "locally homeomorphic?"

I don't need to get too deep into either concept, I just need the correct terminology so I can talk about it unambiguously.