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 $U \in \mathcal N_x$ is automatically a subset of $X$. Kahen (talk) 15:22, 11 May 2016 (UTC)