TITLE: Difference between a "topology" and a "space"?
QUESTION [12 upvotes]: 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?

REPLY [2 votes]: 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 $X$).