# The “standard” definition of discrete subspace

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?

• 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 – H_1317 Mar 22 at 6:08