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- Homework Statement
- Let ##X## be a topological space so that, for any topological space ##Y##, any map ##f:X\longrightarrow Y## is continuous. Prove that ##X## is the discrete topology.

- Relevant Equations
- None

Sketch of proof:

##1.## Let ##V## be open in ##Y##.

##2.## For arbitrary ##f:X\longrightarrow Y## and for arbitrary ##V##, ##f^{-1}(V)## is in ##X##.

##3.## ##f:X\longrightarrow Y## is continuous, so ##f^{-1}(V)## is open in ##X##.

##4.## Every subset ##f^{-1}(V)## of ##X## is open, so ##X## is equipped with the discrete topology.

Question: are "member" and "element" interchangeable terms in mathematics?

Question: Is it necessary to define an open-set cover of ##X## composed of ##f^{-1}(V)##?

##1.## Let ##V## be open in ##Y##.

##2.## For arbitrary ##f:X\longrightarrow Y## and for arbitrary ##V##, ##f^{-1}(V)## is in ##X##.

##3.## ##f:X\longrightarrow Y## is continuous, so ##f^{-1}(V)## is open in ##X##.

##4.## Every subset ##f^{-1}(V)## of ##X## is open, so ##X## is equipped with the discrete topology.

Question: are "member" and "element" interchangeable terms in mathematics?

Question: Is it necessary to define an open-set cover of ##X## composed of ##f^{-1}(V)##?