What does mean by a Topological closure $ F' $ of a field $ F $ with respect to a norm $ ||.|| $?

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What does mean by a Topological closure $ F' $ of a field $ F $ with respect to a norm $ ||.|| $ ?



Answer:



I know the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S.



But i can't explain what is meant by topological closure of a field ?



Can someone help me ?




abstract-algebra algebraic-topology




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  • What is the context here? Is $F$ an arbitrary field or do we know more?
    – Matt
    Aug 16 at 10:57










  • Specifically, the obvious context in which this makes sense is if $F$ is naturally contained within some larger normed field, is this true here?
    – Christopher
    Aug 16 at 10:58










  • I can't find any support for this, but I'm throwing it out to see if it rings true with anyone else. A topology could be said to be closed with respect to a norm if it contains all finite unions and intersections of balls under the metric derived from the norm.
    – Steve B
    Aug 16 at 11:10










  • Here $ F $ is p-adic field ?
    – yourmath
    Aug 16 at 11:27






  • 1




    The only thing that makes sense to me, in that case, is that we're talking about the metric completion. "Topological closure" would be an odd way to refer to that, though.
    – Matt
    Aug 16 at 12:36














up vote
0
down vote

favorite












What does mean by a Topological closure $ F' $ of a field $ F $ with respect to a norm $ ||.|| $ ?



Answer:



I know the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S.



But i can't explain what is meant by topological closure of a field ?



Can someone help me ?




abstract-algebra algebraic-topology




share|cite|improve this question




















  • What is the context here? Is $F$ an arbitrary field or do we know more?
    – Matt
    Aug 16 at 10:57










  • Specifically, the obvious context in which this makes sense is if $F$ is naturally contained within some larger normed field, is this true here?
    – Christopher
    Aug 16 at 10:58










  • I can't find any support for this, but I'm throwing it out to see if it rings true with anyone else. A topology could be said to be closed with respect to a norm if it contains all finite unions and intersections of balls under the metric derived from the norm.
    – Steve B
    Aug 16 at 11:10










  • Here $ F $ is p-adic field ?
    – yourmath
    Aug 16 at 11:27






  • 1




    The only thing that makes sense to me, in that case, is that we're talking about the metric completion. "Topological closure" would be an odd way to refer to that, though.
    – Matt
    Aug 16 at 12:36












up vote
0
down vote

favorite









up vote
0
down vote

favorite











What does mean by a Topological closure $ F' $ of a field $ F $ with respect to a norm $ ||.|| $ ?



Answer:



I know the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S.



But i can't explain what is meant by topological closure of a field ?



Can someone help me ?




abstract-algebra algebraic-topology




share|cite|improve this question












What does mean by a Topological closure $ F' $ of a field $ F $ with respect to a norm $ ||.|| $ ?



Answer:



I know the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S.



But i can't explain what is meant by topological closure of a field ?



Can someone help me ?





abstract-algebra algebraic-topology





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 16 at 10:41









yourmath

1,8291617




1,8291617











  • What is the context here? Is $F$ an arbitrary field or do we know more?
    – Matt
    Aug 16 at 10:57










  • Specifically, the obvious context in which this makes sense is if $F$ is naturally contained within some larger normed field, is this true here?
    – Christopher
    Aug 16 at 10:58










  • I can't find any support for this, but I'm throwing it out to see if it rings true with anyone else. A topology could be said to be closed with respect to a norm if it contains all finite unions and intersections of balls under the metric derived from the norm.
    – Steve B
    Aug 16 at 11:10










  • Here $ F $ is p-adic field ?
    – yourmath
    Aug 16 at 11:27






  • 1




    The only thing that makes sense to me, in that case, is that we're talking about the metric completion. "Topological closure" would be an odd way to refer to that, though.
    – Matt
    Aug 16 at 12:36
















  • What is the context here? Is $F$ an arbitrary field or do we know more?
    – Matt
    Aug 16 at 10:57










  • Specifically, the obvious context in which this makes sense is if $F$ is naturally contained within some larger normed field, is this true here?
    – Christopher
    Aug 16 at 10:58










  • I can't find any support for this, but I'm throwing it out to see if it rings true with anyone else. A topology could be said to be closed with respect to a norm if it contains all finite unions and intersections of balls under the metric derived from the norm.
    – Steve B
    Aug 16 at 11:10










  • Here $ F $ is p-adic field ?
    – yourmath
    Aug 16 at 11:27






  • 1




    The only thing that makes sense to me, in that case, is that we're talking about the metric completion. "Topological closure" would be an odd way to refer to that, though.
    – Matt
    Aug 16 at 12:36















What is the context here? Is $F$ an arbitrary field or do we know more?
– Matt
Aug 16 at 10:57




What is the context here? Is $F$ an arbitrary field or do we know more?
– Matt
Aug 16 at 10:57












Specifically, the obvious context in which this makes sense is if $F$ is naturally contained within some larger normed field, is this true here?
– Christopher
Aug 16 at 10:58




Specifically, the obvious context in which this makes sense is if $F$ is naturally contained within some larger normed field, is this true here?
– Christopher
Aug 16 at 10:58












I can't find any support for this, but I'm throwing it out to see if it rings true with anyone else. A topology could be said to be closed with respect to a norm if it contains all finite unions and intersections of balls under the metric derived from the norm.
– Steve B
Aug 16 at 11:10




I can't find any support for this, but I'm throwing it out to see if it rings true with anyone else. A topology could be said to be closed with respect to a norm if it contains all finite unions and intersections of balls under the metric derived from the norm.
– Steve B
Aug 16 at 11:10












Here $ F $ is p-adic field ?
– yourmath
Aug 16 at 11:27




Here $ F $ is p-adic field ?
– yourmath
Aug 16 at 11:27




1




1




The only thing that makes sense to me, in that case, is that we're talking about the metric completion. "Topological closure" would be an odd way to refer to that, though.
– Matt
Aug 16 at 12:36




The only thing that makes sense to me, in that case, is that we're talking about the metric completion. "Topological closure" would be an odd way to refer to that, though.
– Matt
Aug 16 at 12:36















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