Revisions to Admissible and proper topologies on $C(X,Y)$

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edited Jun 17, 2016 at 14:15

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Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to C$ let $\Lambda(f):B\to C^A$ be defined by $[\Lambda(f)(b)](a) = f(a,b) \in C$.

Let $X, Y$ be topological spaces; we denote by $C(X,Y)$ the collection of continuous maps $f: X\to Y$.

A topology on $C(X,Y)$ is said to be admissible if for any space $Z$ and any continuous map $g: Z\to C(X,Y)$ the map $\Lambda^{-1}(g): Z\times X \to Y$ is continuous (where $Z\times X$ carries the product topology.

Dually, a topology on $C(X,Y)$ is said to be proper if for any space $Z$ and any continuous map $g\in C(Z\times X, Y)$ the map $\Lambda(g): Z\to C(X,Y)$ is continuous.

It turns out that a topology on $C(X,Y)$ is admissible if and only if the evaluation map $\text{eval}: C(X,Y)\times X \to Y$ defined by $(f,x)\mapsto f(x)$ is continuous. (This is not difficult to prove.)

Is there a similar criterion for proper topologies on $C(X,Y)$? (By "similar" I mean a criterion not involving $Z$, and possibly the evaluation map.)

Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to C$ let $\Lambda(f):B\to C^A$ be defined by $[\Lambda(f)(b)](a) = f(a,b) \in C$.

Let $X, Y$ be topological spaces; we denote by $C(X,Y)$ the collection of continuous maps $f: X\to Y$.

A topology on $C(X,Y)$ is said to be admissible if for any space $Z$ and any continuous map $g: Z\to C(X,Y)$ the map $\Lambda^{-1}(g): Z\times X \to Y$ is continuous (where $Z\times X$ carries the product topology.

Dually, a topology on $C(X,Y)$ is said to be proper if for any space $Z$ and any continuous map $g\in C(Z\times X, Y)$ the map $\Lambda(g): Z\to C(X,Y)$ is continuous.

It turns out that a topology on $C(X,Y)$ is admissible if and only if the evaluation map $\text{eval}: C(X,Y)\times X \to Y$ defined by $(f,x)\mapsto f(x)$ is continuous. (This is not difficult to prove.)

Is there a similar criterion for proper topologies on $C(X,Y)$?

Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to C$ let $\Lambda(f):B\to C^A$ be defined by $[\Lambda(f)(b)](a) = f(a,b) \in C$.

Let $X, Y$ be topological spaces; we denote by $C(X,Y)$ the collection of continuous maps $f: X\to Y$.

A topology on $C(X,Y)$ is said to be admissible if for any space $Z$ and any continuous map $g: Z\to C(X,Y)$ the map $\Lambda^{-1}(g): Z\times X \to Y$ is continuous (where $Z\times X$ carries the product topology.

Dually, a topology on $C(X,Y)$ is said to be proper if for any space $Z$ and any continuous map $g\in C(Z\times X, Y)$ the map $\Lambda(g): Z\to C(X,Y)$ is continuous.

It turns out that a topology on $C(X,Y)$ is admissible if and only if the evaluation map $\text{eval}: C(X,Y)\times X \to Y$ defined by $(f,x)\mapsto f(x)$ is continuous. (This is not difficult to prove.)

Is there a similar criterion for proper topologies on $C(X,Y)$? (By "similar" I mean a criterion not involving $Z$, and possibly the evaluation map.)

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edit approved Jun 17, 2016 at 13:44

Martin Sleziak

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Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to C$ let $\Lambda(f):B\to C^A$ be defined by by $[\Lambda(f)(b)](a) = f(a,b) \in C$.

Let $X, Y$ be topological spaces; we denote by $C(X,Y)$ the collection of continuous maps $f: X\to Y$.

A topology on $C(X,Y)$ is said to be admissible if for any space $Z$ and any continouscontinuous map $g: Z\to C(X,Y)$ the map $\Lambda^{-1}(g): Z\times X \to Y$ is continouscontinuous (where $Z\times X$ carries the product topology.

Dually, a topology on $C(X,Y)$ is said to be proper if for any space $Z$ and any continouscontinuous map $g\in C(Z\times X, Y)$ the map $\Lambda(g): Z\to C(X,Y)$ is continouscontinuous.

It turns out that a topology on $C(X,Y)$ is admissible if and only if the evaluation map $\text{eval}: C(X,Y)\times X \to Y$ defined by $(f,x)\mapsto f(x)$ is continuous. (This is not difficult to prove.)

Is there a similar criterion for proper topologies on $C(X,Y)$?

Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to C$ let $\Lambda(f):B\to C^A$ be defined by by $[\Lambda(f)(b)](a) = f(a,b) \in C$.

Let $X, Y$ be topological spaces; we denote by $C(X,Y)$ the collection of continuous maps $f: X\to Y$.

A topology on $C(X,Y)$ is said to be admissible if for any space $Z$ and any continous map $g: Z\to C(X,Y)$ the map $\Lambda^{-1}(g): Z\times X \to Y$ is continous (where $Z\times X$ carries the product topology.

Dually, a topology on $C(X,Y)$ is said to be proper if for any space $Z$ and any continous map $g\in C(Z\times X, Y)$ the map $\Lambda(g): Z\to C(X,Y)$ is continous.