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?
 A: I think of a "space" as the conceptually smallest place in which a given abstraction makes sense. For example, in a metric space, we have distilled the notion of distance. In a topological space, we are in the minimal setting for continuity.  
A: A metric space is a space in which the notion of distance is defined: there is a distance function that exists in that space, and it is true for all points in that space. For example, the distance formula between two points in Euclidean space. 
A topological space is a space in which a notion of "closeness" or "nearness" is understood as existing between some points.
A: 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 $X$).
A: In mathematics, you usually call a set (a collection of objects) with some additional structures a space.  So for example, a set with a certain distance function is called a metric space, and a set with certain subsets defined to be open is called a topological space.  (of course in these two examples the distance function and open sets have to satisfy certain axioms.)  There are tons of other spaces too, vector spaces etc.  
The reason some people refer to a topological space as as having a metric topology, is because a metric space is a specific example of a topological space.  hope that helps.
A: A metric space is a set with a metric.
A topological space is a set with a topology.
Both metric spaces and topologies are useful for defining "continuity." 
Every metric space can be made a topological in a useful way, so that the notion of continuity in metric spaces agrees with the notion of continuity in topological spaces. The reverse is not true - the notion of topology is more general, so not every topological space "comes from" a metric space.
Also, two different metrics on the same set can result in the same topology.
A topology is then a "metric topology" if it arises from converting some metric on the set to a topology.
One concept you have in metric spaces that you don't have in topologies is the notion of "uniform continuity."
