I'm reading "Topics on Continua" by Sergio Macías, and he provides the following notation on page 2:

"If $\varepsilon$ is a positive real number, then the symbol $\mathcal{V}_{\varepsilon}^d(A)$ denotes the $\textit{open ball of radius } \varepsilon \textit{ about } A$. " Here, we are considering a metric space $(X, d)$ and a subset $A$. The problem is that he doesn't actually define what an open ball of radius $\varepsilon$ about $A$ actually is. I couldn't find anywhere online how this is defined. I feel like I should know this by now but honestly I've never seen it in any other classes or text books (or even any papers for that matter). Does anyone know how such an "open ball" is defined when we are centered about a general set instead of just a point? Thanks.

  • $\begingroup$ think in term of neighbors that, within the metric $d$, are less than $\varepsilon$ near. $\endgroup$ – janmarqz Nov 5 '15 at 23:03
  • $\begingroup$ So it would just be the Union of all such open balls? That's what I'd figured, but geometrically that doesn't always make a "ball" in the intuitive sense of the word...not that it should matter since this is topology and because there are various types of metrics that don't always make round shapes when centered on points. I guess I was just a little disturbed by such a notion. $\endgroup$ – gorzardfu Nov 5 '15 at 23:06
  • $\begingroup$ $B_{\varepsilon}(A)=\bigcup_{p\in A}B_{\varepsilon}(p)$ could be? $\endgroup$ – janmarqz Nov 5 '15 at 23:09

It's the set $ \{x \in X \,|\, d(x, A) < \varepsilon\} $, where $d(x, A) = \inf \{d(x, y) \,|\, y \in A\}$.


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