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Topology

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Continuity in a Component or Variable

TOPO-3XNBNY

Which of the following statements are true for any topological spaces $A$, $B$ and $C$?

A

If $f:A\rightarrow B\times C$ where $f(a)=(g(a),h(a))$ is a continuous function then $g:A\rightarrow B$ and $h:A\rightarrow C$ are continuous functions.

B

If $g:A\rightarrow B$ and $h:A\rightarrow C$ are continuous functions then $f:A\rightarrow B\times C$ where $f(a)=(g(a),h(a))$ is a continuous function.

C

If $f:A\times B\rightarrow C$ is a continuous function then for any $a_0\in A$ and $b_0\in B$, $g:A\rightarrow C$ where $g(a)=f(a,b_0)$ and $h:B\rightarrow C$ where $h(b)=f(a_0,b)$ are continuous functions.

D

If for any $a_0\in A$ and $b_0\in B$, $g:A\rightarrow C$ where $g(a)=f(a,b_0)$ and $h:B\rightarrow C$ where $h(b)=f(a_0,b)$ are continuous functions, then $f:A\times B\rightarrow C$ is a continuous function then .