Why do we need separability? In topology a space X is said to be separable if there exist a countable subset of X s.t. it is dense in X.
My question is why it is so important this type of space?
I think that it is important because it acts just like countable sets, even though it can be uncountable, but I'm waiting for more detailed answer.
 A: One nice property is that it controls the cardinality of mapping sets out of the space, making them act more like countable spaces than uncountable. More concretely, suppose $X$ is a separable space and consider the set of all continuous maps $X\to Y$ where $Y$ is some Hausdorff space - we'll call this space $C(X,Y)$.
Now, an upper bound for the cardinality of $C(X,Y)$ is $|Y|^{|X|}$ as this is the cardinality of the set of all function $X\to Y$ which is clearly a superset of $C(X,Y)$. Actually though, we can bring this cardinality down because $X$ is separable. Let $C$ be a countable infinite dense subset of $X$ and recall that if $f\colon X\to Y$ is a continuous map, then $f|_C\colon C\to Y$, the restriction of $f$ to $C$, extends uniquely to a continuous map $X\to Y$ because $C$ is dense and $Y$ is Hausdorff, and in fact it extends to $f$. We say that all continuous maps are determined by their restriction to $C$.
It follows that $|C(X,Y)|$ actually has an upper bound of $|Y|^{|C|}=|Y|^{\aleph_{0}}$. So, suppose $Y$ has the cardinality of the continuum, then $|C(X,Y)|$ has cardinalty at most $$\mathfrak{c}^{\aleph_0}=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}=\mathfrak{c}$$ which is also clearly a lower bound, and so $|C(X,Y)|=\mathfrak{c}$. For instance, one special case is that $|C(\mathbb{R},\mathbb{R})|=\mathfrak{c}<2^{\mathfrak{c}}=|\mathbb{R}|^{|\mathbb{R}|}$ - a surprising result.
