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

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