How do we define total boundedness for a subset $Y$ of metric space $(X,d)$?

Question

How do we define total boundedness for a subset $Y$ of metric space $(X,d)$?

If $(X,d)$ is a metric space, then $X$ is totally bounded if $\forall \epsilon>0 \;\; \exists n(\epsilon) \in N$ such that $\exists x_1,x_2\ldots x_n$ in X such that

$X= \cup_{i=1 }^{n} B_{\epsilon}(x_i)$

Now let $Y\subseteq (X,d)$, then how is total boundedness for $Y$ defined?

the centre of the balls must lie in only $Y$ or can they also lie in $X$ ? Also, the balls should be open in $Y$ or can we have that the balls be open in $X$?

Answer 1

The usual definition (see, for example, here: https://en.wikipedia.org/wiki/Totally_bounded_space ) is that $Y$ is totally bounded in $X$ if there exists a family of subsets of $X$ of arbitrary size that is a finite cover for $Y$. So the answers to your questions are:

1. No; the centres of the ball can lie outside of $Y$ but must still lie in $X$
2. The balls must be open in $X$.