TITLE: Is every manifold a metric space?
QUESTION [24 upvotes]: I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I want to explore the relationship between metric spaces and manifolds, could it be that all manifolds are examples of metric spaces?

REPLY [2 votes]: This is too long for a comment. You will probably think it is pedantic, but I promise it isn't!
In mathematics, we make a distinction between structures and properties. 
A metric space is a structure, and "being a manifold" is a property that a topological space can have. Therefore, the question doesn't make sense. (In practice, some working mathematicians are less careful with this distinction than others!)
More detail:
A topological space is a structure. If someone comes up to you and say

Look at this set. Is it a topological space?

you should reject this question as meaningless. They haven't told you what subsets are supposed to be open.
Similarly,

Is this topological space a metric space?

is a meaningless question. However, the following

(*) Is there a metric on the underlying set of this topological space whose associated topological space is this one?

is a perfectly reasonable question.
A less wordy thing to say is

Is this topological space metrizable?

when we mean (*). Here, metrizable is a property.
A manifold is a certain type of topological space, which is to say, it is a topological space with the property that each  That is, if someone comes up to you and says

Look at this topological space. Is it a manifold?

then in fact they have asked you a reasonable question. The answer is yes if for all points of that space, there exists an open neighborhood homeomorphic to an open subset of $\mathbb{R}^n$ and no otherwise.
So the right question is

Is every manifold metrizable?