 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Mathematics course of the University of Oxford

Year 2 of the mathematics course of the University of Oxford

Topology course of the University of Oxford

Past exam of the topology course of the University of Oxford

2014

Because $f$ is <surjective>, $f^{-1}(B)$ is infinite whenever $B$ is infinite; a finite <preimage> could have only finitely many <image> points. By part (i), its <closure> is $X$. Apply part (a)(ii):
$$
Y=f(X)=f\bigl(\overline{f^{-1}(B)}\bigr)
\subseteq\overline{f(f^{-1}(B))}=\overline B.
$$
The other inclusion is automatic, so
$$
\boxed{\overline B=Y\quad\text{for every infinite }B\subseteq Y.}
$$
Both <continuity> and <surjectivity> enter the argument.

Solution

Past exam of the topology course of the University of Oxford / 2014 / 1 / b / ii / Solution

Past exam of the topology course of the University of Oxford / 2014 / 1 / b / ii / Solution

 Ancestors (13)

 Discussion (0) New discussion

 Articles by others on the same topic (0)

Past exam of the topology course of the University of Oxford / 2014 / 1 / b / ii / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1) New discussion

 Articles by others on the same topic (0)

 Discussion (0) New discussion