quick definition check: locally compact I have this statement in something I am reading:

Let $S$ be a Borel subset of $\mathbb{R}^d$ that is locally compact in
  the relative topology.

Does this mean that every point in $S$ has a compact neighbourhood in the subspace topology of $S$ relative to the usual topology of $\mathbb{R}^d$?
Then it goes on saying

We denote as $B_b(S)$ the linear space of all bounded Borel measurable
  function from $S$ to $\mathbb{R}$Banach space) with respect to the
  supremum norm for each $f\in B_b(S)$.

I have typed every word from the book. Clearly, there is a typo here, does anyone have a guess what it could mean? More importantly, I was wondering why the local compactness condition imposed on $S$ is important.
 A: Summarizing the comments and expanding a little bit:
Yes, your interpretation of "locally compact" is correct.
I think one gets a slightly better feeling what this means by noting that a subset $S$ of $\mathbb{R}^d$ is locally compact in its subspace topology if and only if it is the intersection of an open set with a closed set: $S = F \cap U$ with $F$ closed and $U$ open in $\mathbb R^d$. Indeed, an open set $U$ and a closed set $F$ of $\mathbb{R}^d$ are clearly locally compact and so is their intersection $F \cap U$. Conversely, one can show that $S$ is open in its closure $F = \overline{S}$ in $\mathbb{R}^d$. This means that there is an open set $U \subseteq \mathbb{R}^d$ such that $U \cap F = S$.

It is a general fact that the space $B_{\rm b}(S,\mathcal{F})$ of bounded measurable functions on a measurable space $(S,\mathcal{F})$ is a Banach space with respect to the supremum norm. This has nothing to do with local compactness.
On the other hand, if $S$ is any topological space, then $C_{\rm b}(S) \subseteq B_{\rm b}(S)$ holds. There may or may not be any continuous functions with compact support on $S$, but in any case $C_{c}(S)$ is dense $C_{0}(S)$ which in turn is contained in $C_{\rm b}(S)$. Local compactness of $S$ simply ensures a decent supply of functions in $C_{c}(S)$ and $C_{0}(S)$.

The confusing typo in the quote was introduced in the 2009 edition of the book. The 2004 edition says:

Let $S$ be a Borel subset of $\mathbb R^d$ that is locally compact in the relative topology. We denote as $B_{\rm b}(S)$ the linear space of all bounded Borel measurable functions from $S$ to $\mathbb R$. This becomes a normed space (in fact, a Banach space) with respect to $\lVert f\rVert = \sup_{x\in S} \lvert f(x)\rvert$ for each $f \in B_{\rm b}(S)$.

The missing fragment was probably deleted by accident.
