Timeline for The Tychonov cube $X^X$ of a compact space $X$ is a compact semigroup with the composition operation
Current License: CC BY-SA 4.0
7 events
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Jan 16, 2021 at 7:20 | history | edited | andpe | CC BY-SA 4.0 |
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Jan 16, 2021 at 7:14 | vote | accept | andpe | ||
Jan 15, 2021 at 23:45 | comment | added | Pietro Majer | By definition a map $Y\to X^X$ is continuous iff for any $x\in X$ the composition with the evaluation at $x$ (projection on the $x$-coordinate of the product) is continuous. Thus the continuity of $X^X\ni f\mapsto f\circ g\in X^X$ is obvious, since $X^X\ni f\mapsto (f\circ g)(x)\in X$ is just $f\mapsto f(g(x))$, the evaluation at $g(x)$. | |
Jan 15, 2021 at 14:33 | review | Close votes | |||
Jan 19, 2021 at 20:26 | |||||
Jan 15, 2021 at 13:31 | answer | added | Henno Brandsma | timeline score: 3 | |
Jan 15, 2021 at 7:00 | comment | added | Gregory Arone | If $A, B$ are sets, $X$ is a topological space, and $\alpha\colon A\to B$ is any function then the induced map $X^B\to X^A$, defined by $f\mapsto f\circ \alpha$ is continuous. This follows from the universal property of product. Taking $A=B=X$ you get the result that you want. | |
Jan 15, 2021 at 6:12 | history | asked | andpe | CC BY-SA 4.0 |