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Martin Sleziak
Martin Sleziak
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Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibylitespossibilites, for me it's just important, that I get CW comlexescomplexes at the end) geometric realization of the singular set of the space.

Is the map $C(X,Y)\rightarrow C(\vert Sing(X)\vert,\vert Sing(Y)\vert)$ we get (as $\vert Sing(-)\vert$ is a functor) continuous in the compact-open topology?

Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibylites, for me it's just important, that I get CW comlexes at the end) geometric realization of the singular set of the space.

Is the map $C(X,Y)\rightarrow C(\vert Sing(X)\vert,\vert Sing(Y)\vert)$ we get (as $\vert Sing(-)\vert$ is a functor) continuous in the compact-open topology?

Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibilites, for me it's just important, that I get CW complexes at the end) geometric realization of the singular set of the space.

Is the map $C(X,Y)\rightarrow C(\vert Sing(X)\vert,\vert Sing(Y)\vert)$ we get (as $\vert Sing(-)\vert$ is a functor) continuous in the compact-open topology?

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Alinas
Alinas
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Is the map between mapping spaces, induced by the functor $\vert Sing(-)\vert$ continuous?

Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibylites, for me it's just important, that I get CW comlexes at the end) geometric realization of the singular set of the space.

Is the map $C(X,Y)\rightarrow C(\vert Sing(X)\vert,\vert Sing(Y)\vert)$ we get (as $\vert Sing(-)\vert$ is a functor) continuous in the compact-open topology?