Question

Theorem 7.20: Let (X, T) be a topological space. Then the following are all topological properties the number of elements in X, the number of T-open sets, and having a T-open set containing n elements (for any natural number n

Please prove

Answer 1

Definition: Topological properties are those properties which remain same under homeomorphism ( bijective, continuous, open map).

# If
(Х,Т)
$\rightarrow$
(Y,T)
is a homeomorphism, then it is a bijection from onto . Hence the number of elements in and in are same, and hence the number of elements in is a topological property.

# If $U_{1},U_{2}\ \epsilon \ T$ with $U_{1}\neq U_{2}$ , then $f(U_{1}),f(U_{2})\ \epsilon \ T'$ and $f(U_{1})\neq f(U_{2})$ (since is a bijection). So, there is an injection from into . Similarly, if $V_{1},V_{2}\ \epsilon \ T'$ with $V_{1}\neq V_{2}$ , then $f^{-1}(V_{1}),f^{-1}(V_{2})\ \epsilon \ T$ and $f^{-1}(V_{1})\neq f^{-1}(V_{2})$ (since is a bijection). So, there is an injection from into . Hence, by Bernstein-Schroeder Theorem, there is a bijection from onto . Therefore, the number of - open sets is a topological property.

# If $U\ \epsilon \ T$ has elements, then $f(U)\ \epsilon \ T'$ also has elements(since is a bijection). Hence having a -open set containing elements is a topological property.