# Talk:Space of mappings, topological

From Encyclopedia of Mathematics

## Topology of uniform convergence

The usage of the phrase *topology of uniform convergence* here does not seem to be the same as that of the article Topology of uniform convergence. Richard Pinch (talk) 17:54, 29 December 2016 (CET)

- Yes! And moreover, this is probably an irrecoverable error of the author (Arkhangel'skii); the same error manifests itself in the Russian version. First, the topology of uniform convergence really depends on the uniform structure (not just topology) on Y. Second, even for Y=[0,1] I do not see how to define the topology of uniform convergence that way (no matter what is the family S) when it is different from the compact-open topology (which happens easily when X is not compact). Boris Tsirelson (talk) 22:04, 29 December 2016 (CET)
- Indeed, consider $X=\R$ and $Y=[-1,1]$. If $S$ contains (at least one) non-compact set $A$, then $\mathfrak{T}$ is too strong, since there exists a continuous $f:X\to Y$ such that $f[A]\subseteq(0,1]$ and $ \inf_{x\in A} f(x)=0 $; the set of all continuous $g:X\to Y$ such that $g[A]\subseteq(0,1]$ does not contain $f-\varepsilon$, thus, is not a neighborhood of $f$ in the topology of uniform convergence. And if $S$ contains only compact sets, then $\mathfrak{T}$ is too weak, weaker than (or equal to) the compact-open topology, that is (in this case) the topology of locally uniform convergence. And the phrase "if $X\in S$, then $\mathfrak{T}$ is called..." in the article makes no sense, since it does not specify, what else is included into $S$. I propose to remove all that from the article. Boris Tsirelson (talk) 07:48, 30 December 2016 (CET)
- It seems to be at best a divergent use of the phrase then. I don't see how to repair it either. Richard Pinch (talk) 10:01, 30 December 2016 (CET)
- Now removed. Boris Tsirelson (talk) 12:47, 30 December 2016 (CET)

- It seems to be at best a divergent use of the phrase then. I don't see how to repair it either. Richard Pinch (talk) 10:01, 30 December 2016 (CET)

**How to Cite This Entry:**

Space of mappings, topological.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Space_of_mappings,_topological&oldid=40107