How math spaces are related to each other? Mathematical spaces are quite the most generic stuff we have in math. A space is a set with a structure embeded (which I often read as a "a set with some rule").
Vector spaces are sets with vectorial rules. Metric spaces are the sets where distance is defined. Normed spaces are the sets where the norm is defined. So on, and so forth.
But I have some difficulty seeing how each space is related to another. Is there some map or illustration showing which spaces are contained in others (for the main spaces, as there are hundreds of spaces)? Also, what is the most generic and comprehensive space?
 A: To be honest, I slightly disagree with the comments above. Although it is true that there is no “most generic” and ”comprehensive space”, I am not sure that Category Theory is the answer to your question.
This is due to the fact that Categories are used very differently from “spaces”.
I will give you a not so short overview of the most common kinds of spaces you might find. I will not give formal definitions: for those you can look Wikipedia. I will only give you the idea behind each space.
This list will by no means be complete; no list will ever be. This is also the reason why “diagrams”, as you looked for, do exist, but are very rare: they are either cluttered or incomplete.
Besides, once you are comfortable with the basics, you should be able to find your way through other kinds of spaces.
I think that the first thing you should learn is to distinguish between the kinds of spaces that need also to be vector spaces and the ones which do not. Some of them have written that they are also “vector spaces” in their definitions, while others are “simply sets”, with or without addition and scalar multiplication.

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*Vector Space: a set of elements which can essentially be added together and multiplied by a scalar. I assume you are comfortable with all of the flavours in which they come, real, complex, infinite dimensional…

*Topological Space: a set with a topology. A topology is a family of subsets that allows to define “closeness” or “proximity” between points, in a qualitative, non-quantitative manner.

*Topological Vector Space (TVS, for short): a Vector Space with a topology.

*Metric Space: a set with a metric, i.e. a function that given two points returns a real number which represents the “distance”. Given a metric, it is natural to define balls and it is easy to prove that those form a topology. Thus, every Metric Space is a topological space, but the converse is not true. A Topological Space whose topology could, in principle, be constructed from a metric, is said to be metrizable.

*Normed Space: this is a Vector Space with a Norm, i.e. a function that to each vector assigns a “length”. Given a norm and vector operations, it is easy to construct a metric. Thus, every Normed Space is a Topological Vector Space. The converse is not true.

*Inner Product Space: this is a Vector Space with an inner product, i.e. a generalization of the dot product. Every inner product induces a norm, by simply multiply every vector by itself. Thus every Inner Product Space is a Normed Space. The converse is not true. For instance, the Lebesgue $L^p$, for every $p \in [1 ; \infty ]$ are all Normed Spaces, but only when $p=2$ they can be Inner Product Spaces too.

*Complete Metric Space or Cauchy Space: a metric space (not necessarily a Vector Space), in which every Cauchy sequence converges to a limit still inside the space, and not outside. Intuitively, Cauchy Spaces contain their borders, and so elements are forced to remain inside, even when taking limits.

*Banach Space: a Normed Vector Space which, when viewed as a Metric Space is also complete. Every finite-dimensional Normed Space is Banach. The concept becomes sensible in infinite dimension. If you take $\mathcal C^0$, the space of all continuous functions, with the Lebesgue $L^2$ norm from above, you will find that it is not Banach: i.e., you can take limits of continuous functions and end up with a non-continuous function. It becomes a Banach Space if one chooses to use the so called “uniform norm” (go see the Uniform Limit Theorem).

*Hilbert Space: an Inner Product Space which, when viewed as a Normed Vector Space, is also Banach. The quintessential example is again $L^2$.

*Measure Space: a set with a family of subsets called a $\sigma$-algebra and a measure function, that to many, but not all, subsets assigns a real number which is intuitively the “size”. The most obvious example is the real plane with the area measure.

*Probability Space: a particular kind of Measure Space, in which the measure function is the probability and the subsets represent events.

*Affine Space: it is not a Vector Space, but it is a construction that stems from one. Essentially, vectors here work as in Physics: they can be added together, via the parallelogram rule, but there are free to roam around, and should not be thought as rooted at the origin. Moreover, it is homogeneous, in the sense that there is no notion of origin. This is a white paper. Keep in mind that every Affine Space has an associated Vector Space.

*Euclidean Vector Space: a usually finite dimensional and usually over the  reals Inner Product Space with a few additional rules. Here the Inner Product works closer to our natural geometric intuitions especially regarding angles.

*Euclidean Space: an Affine Space whose associated Vector Space is a Euclidean Vector Space. Here the five axioms of Euclidean Geometry can be verified, and all of the theorems of elementary geometry hold. The notion of reference frame, as it works in Physics, is of paramount importance.

*Topological Manifold: a Topological Space in which every small enough subset resembles $\mathbb R^n$. Think of a cylinder, or a sphere: they locally resemble $\mathbb R^2$, if you consider the tangent planes.

*Differentiable Manifold: a Topological Manifold in which the resemblance is so good that we can say that it locally resembles some finite dimensional Vector Space. Moreover, we can do (Tensor) Calculus on it.

*Smooth Manifold: a Differentiable Manifold in which the transformation between the Manifold itself and the look alike vector spaces is so good that it is given by a smooth function.

*Tangent Space: not a Space by itself, but rather a Vector Space that can be attached to every point of a Smooth or Differentiable Manifold. Intuitively, these are the tangent planes, but I assure you that if you do things very abstractly, without embedding everything in a Euclidean Space, it is a very painful construction.

*Riemannian Manifold or Riemann Space: not to be confused with a Riemann Surface, which is a totally different thing. A smooth manifold in which at each point a Riemannian Metric Tensor is assigned. Essentially, the Tangent Spaces are also Inner Product Spaces. The Riemannian Metric Tensor is basically a fancy dot product. Here lengths, areas, angles and even geodesics can be defined. They also become Measure Spaces, in some sense. A flat Riemannian Manifold is a Euclidean Space. Besides, many different non-Euclidean Geometries can also be defined, by carefully tweaking the Riemannian Metric Tensor.

One easy way to get many more spaces is to relax a bit the hypotheses. For instance, pseudo-Riemannian Manifolds or pre-Hilbert spaces or semi-Normed Spaces. The possibility are endless. The sky is the ceiling.
