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According to the definition of topological space, a topological space is a set with some very general condition on it, for example continuity.

Is a topological space contained in some euclidean space? Is euclidean space enough to describe all kinds of set?

Is there any suggestion about books for reference?

Maybe put the discussion direction of my question to the area of differentiable manifold is a good idea. This is how my question originally start from.

While I am learning differentiable manifold, I have a feeling that all those knowledges are abstract but seems coming from real life. Maybe I should learn some history about how the topic differentiable manifold originally coming from so that I will well know if the topic is from real world or from another abstract subject. As I remember I somewhere heard that it seems to come from industry.

If this opinion is true, then that will solve a problem about how previous people create differentiable manifold thinking about this question. Should I follow this way of thinking to study it.

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    $\begingroup$ Continuity is not a property of topological spaces, but a property of functions between topological spaces. $\endgroup$
    – J.-E. Pin
    Oct 16 '19 at 4:02
  • $\begingroup$ you are right. I vaguely treat the continuity of resulting graph as the continuity of a map. $\endgroup$
    – Hui YI LU
    Oct 16 '19 at 4:08
  • $\begingroup$ You might be interested in learning about topological manifolds, which are topological spaces that locally “look like” Euclidean space. (Note that this is not necessarily the same as being contained / embedded in some Euclidean space.) $\endgroup$ Oct 16 '19 at 4:38
  • $\begingroup$ Yes, this is the point. I am learning differentiable manifolds now. and find if all those objects are in some euclidean space, that will be easier to image and understand the topic. Differentiable manifold should be start from euclidean space. But wether all those manifolds are from $\mathbb{R}^n$ for some $n$, is a question interesting. At least examples I saw are in Euclidean space, e.g. torus, sphere, projective space, Klein bottle. $\endgroup$
    – Hui YI LU
    Oct 16 '19 at 18:38
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Topological spaces are a far-reaching generalization of metric spaces, which in turn are generalizations of Euclidean spaces. There are many topological spaces which are not Euclidean, for example the discrete space in which all subsets are open (and, consequently, closed), or the trivial space, or any metric space where the metric does not arise from an Euclidean structure, like the space of bounded functions on an interval with the sup norm, or $L^p$ spaces of functions with $p\neq 2$.

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  • $\begingroup$ every countable discrete space and every one-point trivial space embed in Euclidean space. $\endgroup$
    – Mirko
    Oct 16 '19 at 17:37
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Concretely: the two-element space with trivial topology, and the long line, are examples of topological spaces that are not subspaces of Euclidean space. Which is to say, there is no $X\subset \mathbb R^n$ such that the subset topology on $X$ is homeomorphic to either of those examples.

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No, not every topological space is contained in a Euclidean space.

One way of seeing this is by showing the existence of topological spaces which do not share the separation axiom that Euclidean spaces have.

All Euclidean space and their subspaces have the property that about any two unequal points $x$ and $y$ in the space, you can find two neighborhoods $U_x$ and $U_y$ of $x$ and $y$ respectively which have empty intersection ($U_x \cap U_y = \emptyset$).

This property is not true for general topological spaces. Here is a simple counterexample: let $X = \{1, 2, 3, ... \}$. Let's declare that a subset $U$ of $X$ is open if and only if either $U = \emptyset, U = X$, or the complement $X \backslash U$ of $U$ in $X$ is finite.

You can check that this makes $X$ into a topological space satisfying the usual axioms: the empty set and $X$ are open, a finite intersection of open sets is open, and an arbitrary union of open sets is open. But $X$ cannot satisfy the separation axiom above that Euclidean spaces and their subspaces have. This is because if $U$ and $V$ are nonempty open sets in $X$, then $U \cap V$ is never empty.

The separation axiom is often called Hausdorff. The topology on $X$ I mentioned is called the cofinite topology.

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