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I saw this on Wikipedia about the hierarchy of spaces: enter image description here

Later in the article, it states that the fundamental building blocks of mathematical spaces are vector spaces and topological spaces. However, it's not clear to me how topological spaces and vector spaces are related, if at all.

Can someone provide an example of a vector space that is not a topological space and vice versa?

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    $\begingroup$ Take any vector space and give it a discrete topology? Also the circle with the induced topology is definitely not a vector space. $\endgroup$ – Jacky Chong Dec 6 '17 at 23:48
  • $\begingroup$ @JackyChong thanks. Can can you expand on this in an answer, where you show this? $\endgroup$ – user408433 Dec 6 '17 at 23:50
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    $\begingroup$ For my first comment read math.stackexchange.com/questions/1413019/… since I don't think I can do better then what Eric said. The second example is obvious. $\endgroup$ – Jacky Chong Dec 6 '17 at 23:59
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Vector spaces and topological spaces are fundamentally different concepts; one is a set with and addition and scalar multiplication, the other a set together with a set of subsets (the open sets) with certain constraints.

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  • $\begingroup$ A vector space is by nature not a topological space. So is your question whether a vector space can or can not be made into a topological space? Maybe such that addition an scalar multiplication are continuous? $\endgroup$ – Reiner Martin Dec 6 '17 at 23:58
  • $\begingroup$ Thanks...I thought it seemed that way, it's not that a vector space is automatically a topological space or vice versa, you need to explicitly add the required structure since it doesn't come for free in the definition of either. $\endgroup$ – user408433 Dec 7 '17 at 1:56
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The diagram you show is demonstrating that any inner product space is a normed vector space, any normed vector space is in turn a metric space, and any metric space is in its turn a topological space.

A topological space is not a vector space because... well, it's just not. It doesn't satisfy the things it's required to satisfy in order for it to be a vector space. I give you for example the so called Sierpinski Space. This is clearly not a vector space.

A vector space is in turn not a topological space unless you define a topology on it. The comment from Jacky is explaining that given any vector space, you could for example give it the discrete topology, thus giving you a vector space which is also a topological space.

In general, however, a vector space isn't a topological space.

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Your question is a little subtler than you think. This hierarchy is about "spaces" that have a geometric structure. The inner product spaces that are an example of all the containing kinds of spaces have geometric structure that's compatible with the vector space axioms. At the top level that means that vector addition and scalar multiplication are continuous functions in the topology defined by the inner product in the spaces at the bottom of the hierarchy.

To address the exact question you asked, let $V$ be any vector space. Then thinking of $V$ as just a set (ignoring the vector space structure), you can equip it with any topology you like.

There are topological spaces that can't be made into vector spaces. Consider a $6$ element set, with the discrete topology. That can't be given a vector space structure - not even over a finite field.

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    $\begingroup$ The field with $5$ elements is a vector space over itself. $\endgroup$ – Somos Dec 7 '17 at 0:41
  • $\begingroup$ @Somos Oops. I'll change the $5$ to $6$. $\endgroup$ – Ethan Bolker Dec 7 '17 at 0:42

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