What I mean with "intuitive": I can handle some formulas, but since I am not a professional mathematician I am not fluent in all the lingo, so I do not know by heart what "second countable" means. If I have to look up all these terms and try to understand them, it takes so much time, that I forget what I was researching in the first place… so basic terminology is appreciated.
It was previously asked whether every manifold is a metric space, but I have to admit, I did not completely understand the answers. Assuming that a manifold is second-countable, the answer is "yes" (I cannot claim I full understood the property "second countable"). My (non-completely) translation of the answer https://math.stackexchange.com/a/1530066/340174 into an intuitive explanation is
I want to find the distance from $x_0$ to y, both of which are elements of the manifold. Since a manifold is locally Euclidean, I can walk a infinitely small way in an "Euclidean" manner. So, I go a small step from $x_0$ to $x_1$ and I calculate the distance I walked, which is possible, because I can just use the Euclidean distance. I walk from $x_1$ to $x_2$ until I reach y and add up all the distances to the total distance. From all the possible paths I take the one that is the shortest and that is my distance.
First question: It seems intuitively obvious to me that the first three conditions of a metric apply to manifold distances, as I described it above. But how do I know that the triangular condition applies as well to the distance over a manifold? Is there an intuitive explanation in the style I tried above?
Originally I would have guessed (without too much thinking) that every metric space is a manifold, but not the other way around. Since the second part is wrong, I would guess that now, that the first part is also wrong. (Otherwise there would be no need to differentiate the two, right?) But what is that so? I can come of with a metric space, like one based on the Levenshtein distance, which is not continuous and my usual impression of manifolds is that they are continuous (since they are supposed to be Euclidean locally). However it seem there are also discrete manifolds (which I do not understand either).
Second question: What is an intuitive explanation, why metric spaces are not necessarily manifolds?
Best Answer
The question itself is a bit misleading; a manifold by itself has no metric and a metric space by itself has no manifold structure.
But it is a fact that every manifold can be endowed with a metric so that the metric-topology coincides with the manifold-topology, and not vice versa. But this involves quite a lot of definitions and is probably not the answer you seek.
So to make the story short, just think of curves in a 2-dimensional plane. They are all metric spaces, as you can measure distance in the Euclidean plane. They are manifold if, at every point, the curve looks locally like a line segment -- no ends, no branching, no space-filling curves, no fractals...
Simple counter-examples are letters A, B, E, F, H, K, P... X, Y... (considered as curves in a Euclidean plane). Because those have at least one point where various line segments meet together. However, D and O are clearly manifolds, and C, I, J manifolds with boundary.