Subset of class of functions equipped with limit

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Take the class of functions:



$$ R_s = phi(s,x) , text for xin Bbb R, s in Bbb R, tag 1$$



where $x$ is the independent variable and $s$ is a continuously varying parameter. This class lives in a bounded space.



A discrete subset of these functions as a finite set can be represented as:



$$ R_s = phi(1,x), phi(2,x), phi(3,x), ldots , phi(n,x) , text for sin Bbb Z^+ tag 2$$



Is there a notion of taking a "functional limit" of $(2)$ as the number of functions in the space goes to infinity so as to equal the cardinality of $ Bbb R $ and so as to equate the set equipped with the limit to $(1)$?



I was discussing this with someone and they suggested using a net:



$ varphi_S : S to Phi $, where $ S $ is a subset of the reals and $ Phi $ is an open set in the topology. This is the right direction, so can someone help me formalize this in concise and correct notation?



Thanks.




general-topology functional-analysis




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  • See my edits to the question for proper MathJax usage.
    – Michael Hardy
    Aug 9 at 2:19














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0
down vote

favorite












Take the class of functions:



$$ R_s = phi(s,x) , text for xin Bbb R, s in Bbb R, tag 1$$



where $x$ is the independent variable and $s$ is a continuously varying parameter. This class lives in a bounded space.



A discrete subset of these functions as a finite set can be represented as:



$$ R_s = phi(1,x), phi(2,x), phi(3,x), ldots , phi(n,x) , text for sin Bbb Z^+ tag 2$$



Is there a notion of taking a "functional limit" of $(2)$ as the number of functions in the space goes to infinity so as to equal the cardinality of $ Bbb R $ and so as to equate the set equipped with the limit to $(1)$?



I was discussing this with someone and they suggested using a net:



$ varphi_S : S to Phi $, where $ S $ is a subset of the reals and $ Phi $ is an open set in the topology. This is the right direction, so can someone help me formalize this in concise and correct notation?



Thanks.




general-topology functional-analysis




share|cite|improve this question






















  • See my edits to the question for proper MathJax usage.
    – Michael Hardy
    Aug 9 at 2:19












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Take the class of functions:



$$ R_s = phi(s,x) , text for xin Bbb R, s in Bbb R, tag 1$$



where $x$ is the independent variable and $s$ is a continuously varying parameter. This class lives in a bounded space.



A discrete subset of these functions as a finite set can be represented as:



$$ R_s = phi(1,x), phi(2,x), phi(3,x), ldots , phi(n,x) , text for sin Bbb Z^+ tag 2$$



Is there a notion of taking a "functional limit" of $(2)$ as the number of functions in the space goes to infinity so as to equal the cardinality of $ Bbb R $ and so as to equate the set equipped with the limit to $(1)$?



I was discussing this with someone and they suggested using a net:



$ varphi_S : S to Phi $, where $ S $ is a subset of the reals and $ Phi $ is an open set in the topology. This is the right direction, so can someone help me formalize this in concise and correct notation?



Thanks.




general-topology functional-analysis




share|cite|improve this question














Take the class of functions:



$$ R_s = phi(s,x) , text for xin Bbb R, s in Bbb R, tag 1$$



where $x$ is the independent variable and $s$ is a continuously varying parameter. This class lives in a bounded space.



A discrete subset of these functions as a finite set can be represented as:



$$ R_s = phi(1,x), phi(2,x), phi(3,x), ldots , phi(n,x) , text for sin Bbb Z^+ tag 2$$



Is there a notion of taking a "functional limit" of $(2)$ as the number of functions in the space goes to infinity so as to equal the cardinality of $ Bbb R $ and so as to equate the set equipped with the limit to $(1)$?



I was discussing this with someone and they suggested using a net:



$ varphi_S : S to Phi $, where $ S $ is a subset of the reals and $ Phi $ is an open set in the topology. This is the right direction, so can someone help me formalize this in concise and correct notation?



Thanks.





general-topology functional-analysis





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edited Aug 9 at 2:19









Michael Hardy

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asked Aug 8 at 21:10









George Thomas

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  • See my edits to the question for proper MathJax usage.
    – Michael Hardy
    Aug 9 at 2:19
















  • See my edits to the question for proper MathJax usage.
    – Michael Hardy
    Aug 9 at 2:19















See my edits to the question for proper MathJax usage.
– Michael Hardy
Aug 9 at 2:19




See my edits to the question for proper MathJax usage.
– Michael Hardy
Aug 9 at 2:19















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