Hmm, losing heart here slightly. So as quick as I can....
Obviously (why?), no
is compact, yet the Heine-Borel Thm states that the closed and bounded
subsets of any
are compact. The proof is quite hard, but uses only what we have defined so far; on current form, I am not anticipating any attempts.(Cowards!!)
Anyway. Let
be topological spaces, and let
be open. Then the function
is said to be
continuous iff the pre-image set
is also open. (If I had thought there was genuine interest out there, I could probably have expressed this a little less compactly)
That is, continuous functions map open sets to open sets (it should be clear from our early definitions that they also map closed sets to closed sets.
And if
is bijective (usual definition) and with a continuous inverse (same definition as above), then this function is called a
homeomorphism. It is the topological equivalent of an isomorphism.
Continuous maps preserve the topological properties we(!) have been "discussing", that is, the continuous image of a connected space is connected, the continuous image of a Hausdorff space is Hausdorff, the continuous image of a compact space is compact etc.
It follows then that homeomorphic spaces are topologically indistinguishable, hence the old joke about a topologist's inability to distinguish her coffee mug from a donut.
I had planned to go from here, or hereabouts, to manifolds, then to Lie groups and their algebra. But I rather think now that I shan't......
