Let $X$ be a topological space and $D\subset X$ a generic subspace. In literature I have found the following different definitions:

  1. $D$ is discrete in $X$ if $X$ doesn't contain limit points of $D$.
  2. $D$ is discrete in $X$ if every point of $D$ is isolated respect to $X$.

Clearly $1.\Rightarrow 2.$, but the converse it is not true. Consider for example $$D=\left\{\frac{1}{n}\,: n\in\mathbb N\, \right\}\subset\mathbb C$$ Every point of $D$ is isolated, but $0$ is a limit point of $D$.

Which of the above definitions is the most used to characterize discrete sets?

  • $\begingroup$ I believe u meant #2 to be “with respect to the subspace topology for D contained in X”. The way it reads now it makes it seem that u are referring to singleton open sets as open in the topology on X $\endgroup$ – H_1317 Mar 22 '19 at 6:08

In my experience as a general and set-theoretic topologist, discrete normally just means that the set is discrete in its relative topology; that’s your second alternative. Sets satisfying the first condition are closed, discrete sets.

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