What claims and what Theorem we can prove using density property in Topology and functional analysis?
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From wikipedia simple definition of dense set defined as ,In topology and related areas of mathematics, A subset A of a topological space X is called dense in X if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X , the notion of density is widely useful in probability and physics and all branche of mathematics particulary Topology and functional analysis , Now my question here is : What claims and what theorem we can prove in topology and functional analysis using density property and why it's important ?
What claims and what Theorem we can prove using density property in Topology ?
functional-analysis algebraic-topology differential-topology density-function
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
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show 1 more comment
$begingroup$
From wikipedia simple definition of dense set defined as ,In topology and related areas of mathematics, A subset A of a topological space X is called dense in X if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X , the notion of density is widely useful in probability and physics and all branche of mathematics particulary Topology and functional analysis , Now my question here is : What claims and what theorem we can prove in topology and functional analysis using density property and why it's important ?
What claims and what Theorem we can prove using density property in Topology ?
functional-analysis algebraic-topology differential-topology density-function
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
$endgroup$
1
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The Baire Category Theorem!
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– dafinguzman
Jan 2 at 17:37
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I'm no expert in topology, hence commenting rather than answering. But perhaps this is useful: the discussion of covering space actions on pp. 72-73 of pi.math.cornell.edu/~hatcher/AT/AT.pdf makes explicit use of a density argument.
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– bounceback
Jan 2 at 17:38
1
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@dafinguzman Isn't the Baire category theorem within Functional Analysis, whereas the OP is concerned primarily with topology?
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– bounceback
Jan 2 at 17:39
1
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This is a very open-ended question, and I don't think it can be given a clear answer unless you narrow your parameters. But, generally speaking, density is useful in large part because given a topological space $X$, there might be a dense subset $S$ of $X$, where the objects in $S$ have nice properties whereas not all the objects in $X$ have such nice properties. We can work indirectly with the not-so-nice objects in $X$ by selecting a sufficiently close approximation in $S$ which has the nice properties we desire.
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– Ben W
Jan 2 at 17:41
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@bounceback As far as I can see, the title asks about both topology and functional analysis.
$endgroup$
– MisterRiemann
Jan 2 at 17:41
|
show 1 more comment
$begingroup$
From wikipedia simple definition of dense set defined as ,In topology and related areas of mathematics, A subset A of a topological space X is called dense in X if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X , the notion of density is widely useful in probability and physics and all branche of mathematics particulary Topology and functional analysis , Now my question here is : What claims and what theorem we can prove in topology and functional analysis using density property and why it's important ?
What claims and what Theorem we can prove using density property in Topology ?
functional-analysis algebraic-topology differential-topology density-function
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
$endgroup$
From wikipedia simple definition of dense set defined as ,In topology and related areas of mathematics, A subset A of a topological space X is called dense in X if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X , the notion of density is widely useful in probability and physics and all branche of mathematics particulary Topology and functional analysis , Now my question here is : What claims and what theorem we can prove in topology and functional analysis using density property and why it's important ?
What claims and what Theorem we can prove using density property in Topology ?
functional-analysis algebraic-topology differential-topology density-function
functional-analysis algebraic-topology differential-topology density-function
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
asked Jan 2 at 17:22
zeraoulia rafikzeraoulia rafik
2,40311029
2,40311029
1
$begingroup$
The Baire Category Theorem!
$endgroup$
– dafinguzman
Jan 2 at 17:37
$begingroup$
I'm no expert in topology, hence commenting rather than answering. But perhaps this is useful: the discussion of covering space actions on pp. 72-73 of pi.math.cornell.edu/~hatcher/AT/AT.pdf makes explicit use of a density argument.
$endgroup$
– bounceback
Jan 2 at 17:38
1
$begingroup$
@dafinguzman Isn't the Baire category theorem within Functional Analysis, whereas the OP is concerned primarily with topology?
$endgroup$
– bounceback
Jan 2 at 17:39
1
$begingroup$
This is a very open-ended question, and I don't think it can be given a clear answer unless you narrow your parameters. But, generally speaking, density is useful in large part because given a topological space $X$, there might be a dense subset $S$ of $X$, where the objects in $S$ have nice properties whereas not all the objects in $X$ have such nice properties. We can work indirectly with the not-so-nice objects in $X$ by selecting a sufficiently close approximation in $S$ which has the nice properties we desire.
$endgroup$
– Ben W
Jan 2 at 17:41
$begingroup$
@bounceback As far as I can see, the title asks about both topology and functional analysis.
$endgroup$
– MisterRiemann
Jan 2 at 17:41
|
show 1 more comment
1
$begingroup$
The Baire Category Theorem!
$endgroup$
– dafinguzman
Jan 2 at 17:37
$begingroup$
I'm no expert in topology, hence commenting rather than answering. But perhaps this is useful: the discussion of covering space actions on pp. 72-73 of pi.math.cornell.edu/~hatcher/AT/AT.pdf makes explicit use of a density argument.
$endgroup$
– bounceback
Jan 2 at 17:38
1
$begingroup$
@dafinguzman Isn't the Baire category theorem within Functional Analysis, whereas the OP is concerned primarily with topology?
$endgroup$
– bounceback
Jan 2 at 17:39
1
$begingroup$
This is a very open-ended question, and I don't think it can be given a clear answer unless you narrow your parameters. But, generally speaking, density is useful in large part because given a topological space $X$, there might be a dense subset $S$ of $X$, where the objects in $S$ have nice properties whereas not all the objects in $X$ have such nice properties. We can work indirectly with the not-so-nice objects in $X$ by selecting a sufficiently close approximation in $S$ which has the nice properties we desire.
$endgroup$
– Ben W
Jan 2 at 17:41
$begingroup$
@bounceback As far as I can see, the title asks about both topology and functional analysis.
$endgroup$
– MisterRiemann
Jan 2 at 17:41
1
1
$begingroup$
The Baire Category Theorem!
$endgroup$
– dafinguzman
Jan 2 at 17:37
$begingroup$
The Baire Category Theorem!
$endgroup$
– dafinguzman
Jan 2 at 17:37
$begingroup$
I'm no expert in topology, hence commenting rather than answering. But perhaps this is useful: the discussion of covering space actions on pp. 72-73 of pi.math.cornell.edu/~hatcher/AT/AT.pdf makes explicit use of a density argument.
$endgroup$
– bounceback
Jan 2 at 17:38
$begingroup$
I'm no expert in topology, hence commenting rather than answering. But perhaps this is useful: the discussion of covering space actions on pp. 72-73 of pi.math.cornell.edu/~hatcher/AT/AT.pdf makes explicit use of a density argument.
$endgroup$
– bounceback
Jan 2 at 17:38
1
1
$begingroup$
@dafinguzman Isn't the Baire category theorem within Functional Analysis, whereas the OP is concerned primarily with topology?
$endgroup$
– bounceback
Jan 2 at 17:39
$begingroup$
@dafinguzman Isn't the Baire category theorem within Functional Analysis, whereas the OP is concerned primarily with topology?
$endgroup$
– bounceback
Jan 2 at 17:39
1
1
$begingroup$
This is a very open-ended question, and I don't think it can be given a clear answer unless you narrow your parameters. But, generally speaking, density is useful in large part because given a topological space $X$, there might be a dense subset $S$ of $X$, where the objects in $S$ have nice properties whereas not all the objects in $X$ have such nice properties. We can work indirectly with the not-so-nice objects in $X$ by selecting a sufficiently close approximation in $S$ which has the nice properties we desire.
$endgroup$
– Ben W
Jan 2 at 17:41
$begingroup$
This is a very open-ended question, and I don't think it can be given a clear answer unless you narrow your parameters. But, generally speaking, density is useful in large part because given a topological space $X$, there might be a dense subset $S$ of $X$, where the objects in $S$ have nice properties whereas not all the objects in $X$ have such nice properties. We can work indirectly with the not-so-nice objects in $X$ by selecting a sufficiently close approximation in $S$ which has the nice properties we desire.
$endgroup$
– Ben W
Jan 2 at 17:41
$begingroup$
@bounceback As far as I can see, the title asks about both topology and functional analysis.
$endgroup$
– MisterRiemann
Jan 2 at 17:41
$begingroup$
@bounceback As far as I can see, the title asks about both topology and functional analysis.
$endgroup$
– MisterRiemann
Jan 2 at 17:41
|
show 1 more comment
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1
$begingroup$
The Baire Category Theorem!
$endgroup$
– dafinguzman
Jan 2 at 17:37
$begingroup$
I'm no expert in topology, hence commenting rather than answering. But perhaps this is useful: the discussion of covering space actions on pp. 72-73 of pi.math.cornell.edu/~hatcher/AT/AT.pdf makes explicit use of a density argument.
$endgroup$
– bounceback
Jan 2 at 17:38
1
$begingroup$
@dafinguzman Isn't the Baire category theorem within Functional Analysis, whereas the OP is concerned primarily with topology?
$endgroup$
– bounceback
Jan 2 at 17:39
1
$begingroup$
This is a very open-ended question, and I don't think it can be given a clear answer unless you narrow your parameters. But, generally speaking, density is useful in large part because given a topological space $X$, there might be a dense subset $S$ of $X$, where the objects in $S$ have nice properties whereas not all the objects in $X$ have such nice properties. We can work indirectly with the not-so-nice objects in $X$ by selecting a sufficiently close approximation in $S$ which has the nice properties we desire.
$endgroup$
– Ben W
Jan 2 at 17:41
$begingroup$
@bounceback As far as I can see, the title asks about both topology and functional analysis.
$endgroup$
– MisterRiemann
Jan 2 at 17:41