Difference between a "topology" and a "space"? What do we mean when we talk about a topological space or a metric space?  I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space?  What is it that the word means, and if there are multiple meanings how can one distinguish them?

Probably the best way to answer your question is to describe an observation: 
Let $X:=(-1,1)$. Define two metrics on $X$ as follows 
$$
d_1(x,y):=|x-y|, \forall x,y\in X,
$$
and
$$
d_2(x,y):=|arctan(x)-arctan(y)|, \forall x,y\in X.
$$
Then you can see that metrics $d_1$ and $d_2$ define the same topology on $X$. However they define different metric structures on $X$. The reason is that $(X,d_1)$ is not a complete metric space (every sequence approaching to 1 is a Cauchy sequence in $(X,d_1)$ but it is not convergent to any point in $X$.), while $(X,d_2)$ is a complete metric space (because it is basically the same as $\mathbb{R}$ with the usual metric which has been shrunk in $X$).
