Prove that there exists a finite subset {$b_1,b_2,…,b_n$} of B such that for all x $in$ K there exists i...
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Let K be a compact metric space. Let B be a dense subset of K. Fix an arbitrary real positive number $delta$. Prove that there exists a finite subset {$b_1,b_2,...,b_n$} of B such that for all x $in$ K there exists i $leq$ n such that $d(x,b_i)<delta$.
This is what I'm thinking for my proof:
Let $K$ be a compact metric space. Let ${B_{delta}(b_i)}_{b_i in B}$ be an open cover of $K$. Thus, by definition, $k subseteq bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. Let $x in K$, then $x in bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. So $x in B_{delta}(b_i)$ for some $i leq n$. Hence for all $x in K$ there exists $i leq n$ such that $d(x,b_i)<delta$.
But I don't make use of the fact that B is a dense subset?
real-analysis
asked Dec 4 at 1:05
big_math_boy
11
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up vote
-1
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favorite
Let K be a compact metric space. Let B be a dense subset of K. Fix an arbitrary real positive number $delta$. Prove that there exists a finite subset {$b_1,b_2,...,b_n$} of B such that for all x $in$ K there exists i $leq$ n such that $d(x,b_i)<delta$.
This is what I'm thinking for my proof:
Let $K$ be a compact metric space. Let ${B_{delta}(b_i)}_{b_i in B}$ be an open cover of $K$. Thus, by definition, $k subseteq bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. Let $x in K$, then $x in bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. So $x in B_{delta}(b_i)$ for some $i leq n$. Hence for all $x in K$ there exists $i leq n$ such that $d(x,b_i)<delta$.
But I don't make use of the fact that B is a dense subset?
real-analysis
asked Dec 4 at 1:05
big_math_boy
11
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let K be a compact metric space. Let B be a dense subset of K. Fix an arbitrary real positive number $delta$. Prove that there exists a finite subset {$b_1,b_2,...,b_n$} of B such that for all x $in$ K there exists i $leq$ n such that $d(x,b_i)<delta$.
This is what I'm thinking for my proof:
Let $K$ be a compact metric space. Let ${B_{delta}(b_i)}_{b_i in B}$ be an open cover of $K$. Thus, by definition, $k subseteq bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. Let $x in K$, then $x in bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. So $x in B_{delta}(b_i)$ for some $i leq n$. Hence for all $x in K$ there exists $i leq n$ such that $d(x,b_i)<delta$.
But I don't make use of the fact that B is a dense subset?
real-analysis
asked Dec 4 at 1:05
big_math_boy
11
Let K be a compact metric space. Let B be a dense subset of K. Fix an arbitrary real positive number $delta$. Prove that there exists a finite subset {$b_1,b_2,...,b_n$} of B such that for all x $in$ K there exists i $leq$ n such that $d(x,b_i)<delta$.
This is what I'm thinking for my proof:
Let $K$ be a compact metric space. Let ${B_{delta}(b_i)}_{b_i in B}$ be an open cover of $K$. Thus, by definition, $k subseteq bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. Let $x in K$, then $x in bigcuplimits_{i=1}^{n} B_{delta}(b_i)$. So $x in B_{delta}(b_i)$ for some $i leq n$. Hence for all $x in K$ there exists $i leq n$ such that $d(x,b_i)<delta$.
But I don't make use of the fact that B is a dense subset?
real-analysis
real-analysis
asked Dec 4 at 1:05
big_math_boy
11
asked Dec 4 at 1:05
big_math_boy
11
edited Dec 7 at 19:40
asked Dec 4 at 1:05
big_math_boy
11
asked Dec 4 at 1:05
big_math_boy
11
asked Dec 4 at 1:05
big_math_boy
11
11
add a comment |
add a comment |
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Hint: Take the union of all $B(b, delta)$, balls of radius $delta$ centered at $b$. This is an open cover of $K$ (why?) and therefore, only finitely many balls cover K.
answered Dec 4 at 1:09
Yunus Syed
1,117218
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Hint: Take the union of all $B(b, delta)$, balls of radius $delta$ centered at $b$. This is an open cover of $K$ (why?) and therefore, only finitely many balls cover K.
answered Dec 4 at 1:09
Yunus Syed
1,117218
add a comment |
up vote
0
down vote
Hint: Take the union of all $B(b, delta)$, balls of radius $delta$ centered at $b$. This is an open cover of $K$ (why?) and therefore, only finitely many balls cover K.
answered Dec 4 at 1:09
Yunus Syed
1,117218
add a comment |
up vote
0
down vote
up vote
0
down vote
Hint: Take the union of all $B(b, delta)$, balls of radius $delta$ centered at $b$. This is an open cover of $K$ (why?) and therefore, only finitely many balls cover K.
answered Dec 4 at 1:09
Yunus Syed
1,117218
Hint: Take the union of all $B(b, delta)$, balls of radius $delta$ centered at $b$. This is an open cover of $K$ (why?) and therefore, only finitely many balls cover K.
answered Dec 4 at 1:09
Yunus Syed
1,117218
answered Dec 4 at 1:09
Yunus Syed
1,117218
answered Dec 4 at 1:09
Yunus Syed
1,117218
answered Dec 4 at 1:09
Yunus Syed
1,117218
1,117218
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