4 1. Topological s
Figure 1
Figure 2
Exercise, a) Prove that the segment [0,1] and the interval (0,1
endowed with the discrete topology are homeomorphic.
b) Prove that the segment and the interval both endowe
the standard topology are not homeomorphic.
c) Are the two objects represented in Figure 2 homeomorp
Separability. A topological space X is said to be a Hausdorff
if any two distinct points x,y G X possess nonintersecting
borhoods. (A neighborhood of a point x is an arbitrary op
containing x.)
1.2. Topological operations on topological spa
The product 1 x 7 . Let X and Y be two topological spaces.
we endow their Cartesian product X xY (i.e.. the set of pairs
where x G X, y G Y) with the standard product topology, a
which consists of Cartesian products of open sets in X and Y
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