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.
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.