Definition talk:Oscillation/Metric Space

I know little about topology.

What is the significance of the statement "Let $X$ be a topological space" in Oscillation at Point? The identification of $X$ as a topological space does not seem to be used anywhere.

Also, the definition of oscillation on a set requires its first argument to be a subset of $X$. Therefore, shouldn't $\left({U; d}\right)$ in $\displaystyle \omega_f \left({x; d}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U; d}\right)$ be changed to $\left({U \cap X; d}\right)$? --Ivar Sand (talk) 10:39, 11 May 2016 (UTC)

Requiring $X$ to be a topological space is necessary because we take the infimum of the oscillations over neighbourhoods of $x \in X$. In particular every $U \in \NN_x$ is automatically a subset of $X$. Kahen (talk) 15:22, 11 May 2016 (UTC)

Thanks, Kahen. Now I understand a bit more. But why is every $U \in \NN_x$ automatically a subset of $X$? What seems to me to be lacking in the definition is a statement expressing an identification or some other kind of relation between the topology $X$ in the definition and the topology $T$ in Definition:Neighborhood of Point in Topological Space. Without such a statement, it seems to me that there is no guarantee that a neighborhood in $\mathcal N_x$ is a subset of $X$. --Ivar Sand (talk) 10:16, 12 May 2016 (UTC)

OK, I assume that it is standard that when a set is turned into a topological space, it is turned into the universal set as well. This is in accordance with what Kahen said and what is said on Definition:Topological Space. --Ivar Sand (talk) 08:11, 13 May 2016 (UTC)

A neighborbood of a point in a topological space (in this case, of $x$ in $X$) is by definition a subset of $X$. $U$ is an element of the set of all neighborhoods of $x$. That is, the infimum is over all the elements of the set of neighborhoods of $x$. That is, the infimum is over all the neighborhoods of $x$ -- each one of which is a subset of $X$.

So what this is saying is: the oscillation of $x$ is the diameter of the "smallest" neighborhood of $x$, where in this case "smallest" means "the one with the least diameter".

I'll tidy this up and make it less confusing (using $X$ both for a set and a topological space is horrible, for a start). --prime mover (talk) 16:31, 3 December 2020 (UTC)