characteristic function of a clopen set continuous?
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Let $X$ be a compact Hausdorff space, $C(X)$ the continuous functions complex valued on $X$ and $Usubset X$ clopen. Is then the characteristic function $1_Uin C(X)$? $1_U:Xtomathbb{C}$ is defined by $1_U(x)=1$, if $xin U$, and zero otherwise.
In general, this function is not continuous. But with $U$ clopen I don't know.
analysis continuity characteristic-functions
asked Nov 16 at 8:30
hetty
1516
add a comment |
up vote
0
down vote
favorite
Let $X$ be a compact Hausdorff space, $C(X)$ the continuous functions complex valued on $X$ and $Usubset X$ clopen. Is then the characteristic function $1_Uin C(X)$? $1_U:Xtomathbb{C}$ is defined by $1_U(x)=1$, if $xin U$, and zero otherwise.
In general, this function is not continuous. But with $U$ clopen I don't know.
analysis continuity characteristic-functions
asked Nov 16 at 8:30
hetty
1516
What is the inverse image of any set?
– Kavi Rama Murthy
Nov 16 at 8:31
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X$ be a compact Hausdorff space, $C(X)$ the continuous functions complex valued on $X$ and $Usubset X$ clopen. Is then the characteristic function $1_Uin C(X)$? $1_U:Xtomathbb{C}$ is defined by $1_U(x)=1$, if $xin U$, and zero otherwise.
In general, this function is not continuous. But with $U$ clopen I don't know.
analysis continuity characteristic-functions
asked Nov 16 at 8:30
hetty
1516
Let $X$ be a compact Hausdorff space, $C(X)$ the continuous functions complex valued on $X$ and $Usubset X$ clopen. Is then the characteristic function $1_Uin C(X)$? $1_U:Xtomathbb{C}$ is defined by $1_U(x)=1$, if $xin U$, and zero otherwise.
In general, this function is not continuous. But with $U$ clopen I don't know.
analysis continuity characteristic-functions
analysis continuity characteristic-functions
asked Nov 16 at 8:30
hetty
1516
asked Nov 16 at 8:30
hetty
1516
asked Nov 16 at 8:30
hetty
1516
asked Nov 16 at 8:30
hetty
1516
asked Nov 16 at 8:30
hetty
1516
1516
What is the inverse image of any set?
– Kavi Rama Murthy
Nov 16 at 8:31
add a comment |
What is the inverse image of any set?
– Kavi Rama Murthy
Nov 16 at 8:31
What is the inverse image of any set?
– Kavi Rama Murthy
Nov 16 at 8:31
What is the inverse image of any set?
– Kavi Rama Murthy
Nov 16 at 8:31
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
If $A$ is any subset of $mathbb{C}$ then
$(1_U)^{-1}[A] = U$ iff $1 in A, 0 notin A$,
$(1_U)^{-1}[A] = Xsetminus U$ if $0 in A, 1 notin A$,
$(1_U)^{-1}[A] = X$ iff ${0,1} subseteq A$, and
$(1_U)^{-1}[A] = emptyset$ if ${0,1} cap A = emptyset$
So for all (open) $A$ the inverse image is open (as $U$ is clopen).
So $1_U$ is continuous.
answered Nov 16 at 8:38
Henno Brandsma
101k344107
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
If $A$ is any subset of $mathbb{C}$ then
$(1_U)^{-1}[A] = U$ iff $1 in A, 0 notin A$,
$(1_U)^{-1}[A] = Xsetminus U$ if $0 in A, 1 notin A$,
$(1_U)^{-1}[A] = X$ iff ${0,1} subseteq A$, and
$(1_U)^{-1}[A] = emptyset$ if ${0,1} cap A = emptyset$
So for all (open) $A$ the inverse image is open (as $U$ is clopen).
So $1_U$ is continuous.
answered Nov 16 at 8:38
Henno Brandsma
101k344107
add a comment |
up vote
3
down vote
accepted
If $A$ is any subset of $mathbb{C}$ then
$(1_U)^{-1}[A] = U$ iff $1 in A, 0 notin A$,
$(1_U)^{-1}[A] = Xsetminus U$ if $0 in A, 1 notin A$,
$(1_U)^{-1}[A] = X$ iff ${0,1} subseteq A$, and
$(1_U)^{-1}[A] = emptyset$ if ${0,1} cap A = emptyset$
So for all (open) $A$ the inverse image is open (as $U$ is clopen).
So $1_U$ is continuous.
answered Nov 16 at 8:38
Henno Brandsma
101k344107
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
If $A$ is any subset of $mathbb{C}$ then
$(1_U)^{-1}[A] = U$ iff $1 in A, 0 notin A$,
$(1_U)^{-1}[A] = Xsetminus U$ if $0 in A, 1 notin A$,
$(1_U)^{-1}[A] = X$ iff ${0,1} subseteq A$, and
$(1_U)^{-1}[A] = emptyset$ if ${0,1} cap A = emptyset$
So for all (open) $A$ the inverse image is open (as $U$ is clopen).
So $1_U$ is continuous.
answered Nov 16 at 8:38
Henno Brandsma
101k344107
If $A$ is any subset of $mathbb{C}$ then
$(1_U)^{-1}[A] = U$ iff $1 in A, 0 notin A$,
$(1_U)^{-1}[A] = Xsetminus U$ if $0 in A, 1 notin A$,
$(1_U)^{-1}[A] = X$ iff ${0,1} subseteq A$, and
$(1_U)^{-1}[A] = emptyset$ if ${0,1} cap A = emptyset$
So for all (open) $A$ the inverse image is open (as $U$ is clopen).
So $1_U$ is continuous.
answered Nov 16 at 8:38
Henno Brandsma
101k344107
answered Nov 16 at 8:38
Henno Brandsma
101k344107
answered Nov 16 at 8:38
Henno Brandsma
101k344107
answered Nov 16 at 8:38
Henno Brandsma
101k344107
101k344107
add a comment |
add a comment |
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What is the inverse image of any set?
– Kavi Rama Murthy
Nov 16 at 8:31