What sort of textbook do I need to read, if I want to learn what "separable metric space" means?

Question

I need to learn what makes a metric space "separable". I have a book on topology that I really like: Crossley's Essential Topology, but it doesn't talk much about it. I have a book on Real analysis that I really like, Abbott's Understanding Real Analysis, but it doesn't talk about it either. I have a book on topological manifolds, Lee's Introduction to Topological Manifolds, and it mentions separability, but in more generalized contexts.

Browsing some questions on Math.SE (e.g. Definition of a separable metric space), it seems that I need to understand concepts like "countable" and "dense", in order to make sense of "separable".

So, what sort of textbook would such notions be covered under, in less generalized contexts than the study of topological manifolds? Please do not suggest Rudin, as I am not mathematically mature enough for the terse nature of his excellent account.

Answer 1

You don't really need to talk about manifolds. A space is separable if there exists a countable, dense subset. You can find any of those definitions in any introductory topology books. You can probably find Munkres "Topology" if you google it.

A countable, dense subset is something you can write as an infinite list of points such that every point in the set is contained in the list or is arbitrarily close to a point in the list. The real numbers are separable because the rationals are a countable, dense subset.

Answer 2

That's a pretty basic concept found in pretty much any introductory text on topology. Introduction to Topology by Gamelin and Greene is available in a Dover paperback edition. A set, A, is "dense" in a topological space, X, if and only if the closure of A contains X. A topological space, X, is "separable" if and only if there exist a [b]countable[/b] set that is dense in X.

Answer 3

There are many texts on introductory topology that cover this. Engelking's 800+page General Topology has 8 chapters; one is entirely on metric spaces.

In the context of metric spaces: When $d$ is a metric on a set $X$ and $Y\subset X$, we say $Y$ is dense in $X$ iff for every $p\in X$ there is a sequence $(q_n)_{n\in \mathbb N}$ of (not necessarily distinct) members of $Y$ such that $\lim_{n\to \infty}d(q_n,p)=0.$ If $p\in Y$ we can take $q_n=p$ for every $n.$ So in particular , $X$ is dense in $X.$

For any $Y\subset X,$ the set of all $p\in X$ for which there exists a sequence $(q_n)_{n\in \mathbb N}$ of members of $Y$ such that $\lim_{n\to \infty}d(q_n,p)=0$ is called the closure of $Y,$ denoted $\bar Y.$ When $p\in Y$ we can take $q_n=p$ for every $n.$ So $Y\subset \overline {Y}.$ From the previous paragraph we see that "$Y$ is dense in $X$" is equivalent to $\bar Y=X.$

A topological space $X$ is separable iff there exists a countable $Y\subset X$ such that $\bar Y=X.$ In other words, $X$ has a countable dense subset. Countable means not uncountable: $Y$ is countable iff there is a surjective function $f:\mathbb N\to Y.$ In particular , finite sets are countable. Countable non-finite sets are usually called countably infinite.

Example. Let $X=\mathbb R$ and $d(x,y)=|x-y|$ for $x,y\in \mathbb R$. Then $X$ is uncountable, but separable. For example, $Y=\mathbb Q$ is countable and $\bar Y=X.$

Separable metric spaces have many useful properties. And almost all applications of analysis in science use separable metric spaces.

Topological spaces form a much bigger class than metric spaces. In many topological spaces, closure and density cannot be described in terms of sequences.