Non-trivial explicit example of a partition of unity
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Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?
general-topology examples-counterexamples paracompactness
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
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add a comment |
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Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?
general-topology examples-counterexamples paracompactness
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
$endgroup$
add a comment |
$begingroup$
Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?
general-topology examples-counterexamples paracompactness
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
$endgroup$
Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?
general-topology examples-counterexamples paracompactness
general-topology examples-counterexamples paracompactness
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
edited Dec 9 '18 at 21:55
Eric Wofsey
186k14215342
186k14215342
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
asked May 18 '16 at 6:47
MonsieurGaloisMonsieurGalois
3,4381333
3,4381333
add a comment |
add a comment |
2 Answers
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Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $mathbb{R}$ given by $U = (-infty,2)$ and $V = (-2,infty)$. A partition of unity associated to ${U,V}$ could be given by ${ f_U, f_V }$, where
$$ f_U, f_V colon mathbb{R} to [0,1]$$
$$ f_U (x) = left{ begin{array}{ll}
1 & text{ if } x leq -1 \
frac{1-x}{2} & text{ if } -1 leq x leq 1 \
0 & text{ if } x geq 1 end{array} right. $$
$$ f_V (x) = left{ begin{array}{ll}
0 & text{ if } x leq -1 \
frac{x-1}{2} & text{ if } -1 leq x leq 1 \
1 & text{ if } x geq 1 end{array} right. $$
Then it is clear that $f_U(x)+f_V(x) = 1$ for all $x$. The support of $f_U$ is contained in $U$ and the support of $f_V$ in $V$. Of course the condition that any $x$ has an open neighbourhood where only a finite number of the functions are different from cero is automatic here because there are only two functions.
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
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I do not know what your precise definition is of a partition of unity, but I like the example of $X=mathbf R$, with open covering ${mathbf Rsetminus pimathbf Z,mathbf Rsetminus (fracpi2+pimathbf Z)}$, and functions $sin^2$ and $cos^2$.
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
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2 Answers
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active
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2 Answers
2
active
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active
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$begingroup$
Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $mathbb{R}$ given by $U = (-infty,2)$ and $V = (-2,infty)$. A partition of unity associated to ${U,V}$ could be given by ${ f_U, f_V }$, where
$$ f_U, f_V colon mathbb{R} to [0,1]$$
$$ f_U (x) = left{ begin{array}{ll}
1 & text{ if } x leq -1 \
frac{1-x}{2} & text{ if } -1 leq x leq 1 \
0 & text{ if } x geq 1 end{array} right. $$
$$ f_V (x) = left{ begin{array}{ll}
0 & text{ if } x leq -1 \
frac{x-1}{2} & text{ if } -1 leq x leq 1 \
1 & text{ if } x geq 1 end{array} right. $$
Then it is clear that $f_U(x)+f_V(x) = 1$ for all $x$. The support of $f_U$ is contained in $U$ and the support of $f_V$ in $V$. Of course the condition that any $x$ has an open neighbourhood where only a finite number of the functions are different from cero is automatic here because there are only two functions.
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
$endgroup$
add a comment |
$begingroup$
Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $mathbb{R}$ given by $U = (-infty,2)$ and $V = (-2,infty)$. A partition of unity associated to ${U,V}$ could be given by ${ f_U, f_V }$, where
$$ f_U, f_V colon mathbb{R} to [0,1]$$
$$ f_U (x) = left{ begin{array}{ll}
1 & text{ if } x leq -1 \
frac{1-x}{2} & text{ if } -1 leq x leq 1 \
0 & text{ if } x geq 1 end{array} right. $$
$$ f_V (x) = left{ begin{array}{ll}
0 & text{ if } x leq -1 \
frac{x-1}{2} & text{ if } -1 leq x leq 1 \
1 & text{ if } x geq 1 end{array} right. $$
Then it is clear that $f_U(x)+f_V(x) = 1$ for all $x$. The support of $f_U$ is contained in $U$ and the support of $f_V$ in $V$. Of course the condition that any $x$ has an open neighbourhood where only a finite number of the functions are different from cero is automatic here because there are only two functions.
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
$endgroup$
add a comment |
$begingroup$
Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $mathbb{R}$ given by $U = (-infty,2)$ and $V = (-2,infty)$. A partition of unity associated to ${U,V}$ could be given by ${ f_U, f_V }$, where
$$ f_U, f_V colon mathbb{R} to [0,1]$$
$$ f_U (x) = left{ begin{array}{ll}
1 & text{ if } x leq -1 \
frac{1-x}{2} & text{ if } -1 leq x leq 1 \
0 & text{ if } x geq 1 end{array} right. $$
$$ f_V (x) = left{ begin{array}{ll}
0 & text{ if } x leq -1 \
frac{x-1}{2} & text{ if } -1 leq x leq 1 \
1 & text{ if } x geq 1 end{array} right. $$
Then it is clear that $f_U(x)+f_V(x) = 1$ for all $x$. The support of $f_U$ is contained in $U$ and the support of $f_V$ in $V$. Of course the condition that any $x$ has an open neighbourhood where only a finite number of the functions are different from cero is automatic here because there are only two functions.
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
$endgroup$
Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $mathbb{R}$ given by $U = (-infty,2)$ and $V = (-2,infty)$. A partition of unity associated to ${U,V}$ could be given by ${ f_U, f_V }$, where
$$ f_U, f_V colon mathbb{R} to [0,1]$$
$$ f_U (x) = left{ begin{array}{ll}
1 & text{ if } x leq -1 \
frac{1-x}{2} & text{ if } -1 leq x leq 1 \
0 & text{ if } x geq 1 end{array} right. $$
$$ f_V (x) = left{ begin{array}{ll}
0 & text{ if } x leq -1 \
frac{x-1}{2} & text{ if } -1 leq x leq 1 \
1 & text{ if } x geq 1 end{array} right. $$
Then it is clear that $f_U(x)+f_V(x) = 1$ for all $x$. The support of $f_U$ is contained in $U$ and the support of $f_V$ in $V$. Of course the condition that any $x$ has an open neighbourhood where only a finite number of the functions are different from cero is automatic here because there are only two functions.
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
answered May 22 '16 at 17:01
Goa'uldGoa'uld
1,901318
1,901318
add a comment |
add a comment |
$begingroup$
I do not know what your precise definition is of a partition of unity, but I like the example of $X=mathbf R$, with open covering ${mathbf Rsetminus pimathbf Z,mathbf Rsetminus (fracpi2+pimathbf Z)}$, and functions $sin^2$ and $cos^2$.
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
$endgroup$
add a comment |
$begingroup$
I do not know what your precise definition is of a partition of unity, but I like the example of $X=mathbf R$, with open covering ${mathbf Rsetminus pimathbf Z,mathbf Rsetminus (fracpi2+pimathbf Z)}$, and functions $sin^2$ and $cos^2$.
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
$endgroup$
add a comment |
$begingroup$
I do not know what your precise definition is of a partition of unity, but I like the example of $X=mathbf R$, with open covering ${mathbf Rsetminus pimathbf Z,mathbf Rsetminus (fracpi2+pimathbf Z)}$, and functions $sin^2$ and $cos^2$.
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
$endgroup$
I do not know what your precise definition is of a partition of unity, but I like the example of $X=mathbf R$, with open covering ${mathbf Rsetminus pimathbf Z,mathbf Rsetminus (fracpi2+pimathbf Z)}$, and functions $sin^2$ and $cos^2$.
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
answered May 18 '16 at 20:39
Johannes HuismanJohannes Huisman
3,125617
3,125617
add a comment |
add a comment |
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