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This problem confuses me for a long time since I am not a student in mathematics. And also it appears quite often here.

Ex:

  1. Image of open set through linear map (please see the comment)
    "Let $X$ be a normed vector space." "Let $X$ be a topological vector
    space."
  2. The preimage of continuous function on a closed set is closed.
    (Please see the last answer)
    "This answer works on metric space." "This answer does not work on topological space."

I know the definition of these nomenclature; however, how to ask a strict problem such that one can understand without confusion still confuses me.

I hope the answer could be categorized and be a big picture.

Specifically, for example

  1. If you want to prove it from the point of view of topology, you have to consider the limit.
  2. When you want to prove it from the point of view of metric space, you have to define the distance of vectors in sequence.
Community
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sleeve chen
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    Wait, what? You know the formal definition, but don't know how to define it? I'm clearly missing something. – Stefan Mesken Aug 04 '16 at 23:16
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    I fail to see what the question is here. – Aweygan Aug 04 '16 at 23:18
  • It is not only how to define it. It is how to view a problem from different point of views. And I cannot clearly identify them. Please see these examples I provide. – sleeve chen Aug 04 '16 at 23:21
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    Are you asking what the properties of real vector spaces are and what types of spaces they generalize to? I.e. real vector spaces have distances, so they generalize to metric spaces, when limits exist, they are unique, generalizes to Hausdorff topological spaces? I really want to help with you r question, but it is also unclear to me what you are asking. – Chill2Macht Aug 05 '16 at 00:54
  • @William Yes. Specifically, like the second example, the one asking question provides an answer holding in metric space; however it may not hold in topological space. Then what should I do further to let it hold in the topological space,which is more general. – sleeve chen Aug 05 '16 at 01:03
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    Well, not every result that holds in a metric space holds in a topological space; otherwise topological spaces wouldn't be more general, they would be equivalent. I agree it can be difficult trying to remember and recognize what is the most general setting in which certain properties hold in. To be honest, I am fairly certain that I do not know the answer for most or all properties, and that even for the ones I think I do, there are probably ways to phrase the setting so that the situation becomes even more general. What I do is try to make sure I at least always know at least one setting for – Chill2Macht Aug 05 '16 at 01:08
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    Limits only hold in metric spaces (think about the delta-epsilon definition and this is obvious) so when working with a topology without a metric defined on it, you can only use the properties of open sets for your proofs. Like I said before, norm induces a metric and a metric induces a topology but the other direction is NOT true. – Wavelet Aug 05 '16 at 01:08
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    which the property is true, even if I don't know the most general setting. There are definitely cases where you can get a precise answer, but sometimes it just leads to forcing yourself to read ncatlab articles you don't understand at all and hating yourself for not knowing more math. Sometimes the most pragmatic option really is just to take it a step at a time and accept that you won't always know the most general possible setting in which a certain property holds. It can also be confusing trying to put all of the pieces back together, although that's not a reason not to try. – Chill2Macht Aug 05 '16 at 01:10
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    Well technically you can define limits in any topological space, and those limits are guaranteed to be unique in at least Hausdorff spaces. There are at least two ways to generalize limits to topological spaces which, to the best of my limited knowledge, might be equivalent: nets and filters. Also you can define notions of convergence in even more general settings than topological spaces, see for example here: https://ncatlab.org/nlab/show/convergence+space Of course, a lot of the nice properties of sequences one takes for granted come from first-countability of a topological space. – Chill2Macht Aug 05 '16 at 01:12
  • @William Good point..I guess I am a little rusty on topology – Wavelet Aug 05 '16 at 01:35

1 Answers1

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A norm induces a metric and a metric induces a topology.

Wavelet
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  • This answer makes me feel better. However, to my second example, if I have a proof constructed in metric space, how should I generalize this proof to topological space? The proof given in the second example shows the concept of limitation. – sleeve chen Aug 04 '16 at 23:40
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    Well I'll give you a hint: taking pre-images commutes with taking compliments in the sense that $f^{-1}(E^{c}) = (f^{-1}(E))^{c}$ – Wavelet Aug 04 '16 at 23:58