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 40 mins ago
hetty
1426
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 40 mins ago
hetty
1426
What is the inverse image of any set?
– Kavi Rama Murthy
39 mins ago
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 40 mins ago
hetty
1426
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 40 mins ago
hetty
1426
asked 40 mins ago
hetty
1426
asked 40 mins ago
hetty
1426
asked 40 mins ago
hetty
1426
asked 40 mins ago
hetty
1426
1426
What is the inverse image of any set?
– Kavi Rama Murthy
39 mins ago
add a comment |
What is the inverse image of any set?
– Kavi Rama Murthy
39 mins ago
What is the inverse image of any set?
– Kavi Rama Murthy
39 mins ago
What is the inverse image of any set?
– Kavi Rama Murthy
39 mins ago
add a comment |
1 Answer
1
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0
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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 32 mins ago
Henno Brandsma
100k344107
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
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 32 mins ago
Henno Brandsma
100k344107
add a comment |
up vote
0
down vote
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 32 mins ago
Henno Brandsma
100k344107
add a comment |
up vote
0
down vote
up vote
0
down vote
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 32 mins ago
Henno Brandsma
100k344107
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 32 mins ago
Henno Brandsma
100k344107
answered 32 mins ago
Henno Brandsma
100k344107
answered 32 mins ago
Henno Brandsma
100k344107
answered 32 mins ago
Henno Brandsma
100k344107
100k344107
add a comment |
add a comment |
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What is the inverse image of any set?
– Kavi Rama Murthy
39 mins ago