Mixing
Showing a proper inclusion
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Walter Rudin, Principles of Mathematical, Question 2.7b.
Let $A_1, A_2,...$ be subsets of a metric space. If $B = cup_{i=1}^infty A_i$, prove that $cup_{i=1}^infty bar{A_i}subset bar{B} $, where $bar{A}$ is the closure of $A$.
Most of the answer(s) provided online proves only inclusion, not proper inclusion (i.e. $cup_{i=1}^infty bar{A_i}subseteq bar{B}$, and not $cup_{i=1}^infty bar{A_i}subset bar{B}$). Could anyone explain why proper inclusion in this case follows immediately from inclusion?
Edit: not a duplicate of (Prove that $bigcup^{infty}_{k=1}overline{A_k} subset bar{B}$ if $B=bigcup^{infty}_{k=1}A_k $), but a case of ambiguity with the notation - Rudin uses $subset$ in place of $subseteq$ (Thanks Alberto!)
real-analysis notation
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
$endgroup$
add a comment |
$begingroup$
Walter Rudin, Principles of Mathematical, Question 2.7b.
Let $A_1, A_2,...$ be subsets of a metric space. If $B = cup_{i=1}^infty A_i$, prove that $cup_{i=1}^infty bar{A_i}subset bar{B} $, where $bar{A}$ is the closure of $A$.
Most of the answer(s) provided online proves only inclusion, not proper inclusion (i.e. $cup_{i=1}^infty bar{A_i}subseteq bar{B}$, and not $cup_{i=1}^infty bar{A_i}subset bar{B}$). Could anyone explain why proper inclusion in this case follows immediately from inclusion?
Edit: not a duplicate of (Prove that $bigcup^{infty}_{k=1}overline{A_k} subset bar{B}$ if $B=bigcup^{infty}_{k=1}A_k $), but a case of ambiguity with the notation - Rudin uses $subset$ in place of $subseteq$ (Thanks Alberto!)
real-analysis notation
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
$endgroup$
1
$begingroup$
Rudin uses $subset$ and $subseteq$ interchangeably.
$endgroup$
– Alberto Takase
Feb 3 at 4:26
$begingroup$
@AlbertoTakase that would explain quite a bit, thanks!
$endgroup$
– Sean Lee
Feb 3 at 4:27
1
$begingroup$
@AlbertoTakase he does not use $subseteq$ at all, see page 3 of his book. He even introduces $in$ as a notation there.
$endgroup$
– Henno Brandsma
Feb 3 at 8:47
$begingroup$
"Usually a mathematician..." would be a more accurate replacement of "Rudin"
$endgroup$
– Alberto Takase
Feb 3 at 9:07
add a comment |
$begingroup$
Walter Rudin, Principles of Mathematical, Question 2.7b.
Let $A_1, A_2,...$ be subsets of a metric space. If $B = cup_{i=1}^infty A_i$, prove that $cup_{i=1}^infty bar{A_i}subset bar{B} $, where $bar{A}$ is the closure of $A$.
Most of the answer(s) provided online proves only inclusion, not proper inclusion (i.e. $cup_{i=1}^infty bar{A_i}subseteq bar{B}$, and not $cup_{i=1}^infty bar{A_i}subset bar{B}$). Could anyone explain why proper inclusion in this case follows immediately from inclusion?
Edit: not a duplicate of (Prove that $bigcup^{infty}_{k=1}overline{A_k} subset bar{B}$ if $B=bigcup^{infty}_{k=1}A_k $), but a case of ambiguity with the notation - Rudin uses $subset$ in place of $subseteq$ (Thanks Alberto!)
real-analysis notation
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
$endgroup$
Walter Rudin, Principles of Mathematical, Question 2.7b.
Let $A_1, A_2,...$ be subsets of a metric space. If $B = cup_{i=1}^infty A_i$, prove that $cup_{i=1}^infty bar{A_i}subset bar{B} $, where $bar{A}$ is the closure of $A$.
Most of the answer(s) provided online proves only inclusion, not proper inclusion (i.e. $cup_{i=1}^infty bar{A_i}subseteq bar{B}$, and not $cup_{i=1}^infty bar{A_i}subset bar{B}$). Could anyone explain why proper inclusion in this case follows immediately from inclusion?
Edit: not a duplicate of (Prove that $bigcup^{infty}_{k=1}overline{A_k} subset bar{B}$ if $B=bigcup^{infty}_{k=1}A_k $), but a case of ambiguity with the notation - Rudin uses $subset$ in place of $subseteq$ (Thanks Alberto!)
real-analysis notation
real-analysis notation
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
edited Feb 12 at 15:34
Sean Lee
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
asked Feb 3 at 4:18
Sean LeeSean Lee
830214
830214
1
$begingroup$
Rudin uses $subset$ and $subseteq$ interchangeably.
$endgroup$
– Alberto Takase
Feb 3 at 4:26
$begingroup$
@AlbertoTakase that would explain quite a bit, thanks!
$endgroup$
– Sean Lee
Feb 3 at 4:27
1
$begingroup$
@AlbertoTakase he does not use $subseteq$ at all, see page 3 of his book. He even introduces $in$ as a notation there.
$endgroup$
– Henno Brandsma
Feb 3 at 8:47
$begingroup$
"Usually a mathematician..." would be a more accurate replacement of "Rudin"
$endgroup$
– Alberto Takase
Feb 3 at 9:07
add a comment |
1
$begingroup$
Rudin uses $subset$ and $subseteq$ interchangeably.
$endgroup$
– Alberto Takase
Feb 3 at 4:26
$begingroup$
@AlbertoTakase that would explain quite a bit, thanks!
$endgroup$
– Sean Lee
Feb 3 at 4:27
1
$begingroup$
@AlbertoTakase he does not use $subseteq$ at all, see page 3 of his book. He even introduces $in$ as a notation there.
$endgroup$
– Henno Brandsma
Feb 3 at 8:47
$begingroup$
"Usually a mathematician..." would be a more accurate replacement of "Rudin"
$endgroup$
– Alberto Takase
Feb 3 at 9:07
1
1
$begingroup$
Rudin uses $subset$ and $subseteq$ interchangeably.
$endgroup$
– Alberto Takase
Feb 3 at 4:26
$begingroup$
Rudin uses $subset$ and $subseteq$ interchangeably.
$endgroup$
– Alberto Takase
Feb 3 at 4:26
$begingroup$
@AlbertoTakase that would explain quite a bit, thanks!
$endgroup$
– Sean Lee
Feb 3 at 4:27
$begingroup$
@AlbertoTakase that would explain quite a bit, thanks!
$endgroup$
– Sean Lee
Feb 3 at 4:27
1
1
$begingroup$
@AlbertoTakase he does not use $subseteq$ at all, see page 3 of his book. He even introduces $in$ as a notation there.
$endgroup$
– Henno Brandsma
Feb 3 at 8:47
$begingroup$
@AlbertoTakase he does not use $subseteq$ at all, see page 3 of his book. He even introduces $in$ as a notation there.
$endgroup$
– Henno Brandsma
Feb 3 at 8:47
$begingroup$
"Usually a mathematician..." would be a more accurate replacement of "Rudin"
$endgroup$
– Alberto Takase
Feb 3 at 9:07
$begingroup$
"Usually a mathematician..." would be a more accurate replacement of "Rudin"
$endgroup$
– Alberto Takase
Feb 3 at 9:07
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$subset$ just means "subset of" in Rudin, not "proper subset of". He does not use $subseteq$. See page 3 in chapter 1 where he introduces this notation.
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
+1: I spent half an hour to see if Rudin uses $subseteq$ to no avail.
$endgroup$
– Alberto Takase
Feb 3 at 9:05
$begingroup$
@AlbertoTakase some books have an index of notations and many have an introductory chapter like Rudin to settle on notations. Too bad people skip them.
$endgroup$
– Henno Brandsma
Feb 3 at 9:13
$begingroup$
I looked at books and published papers; he was consistent. It seems $subseteq$ is modern, but even recently authors (e.g. Munkres) favor $subset$.
$endgroup$
– Alberto Takase
Feb 3 at 9:20
add a comment |
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1 Answer
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$begingroup$
$subset$ just means "subset of" in Rudin, not "proper subset of". He does not use $subseteq$. See page 3 in chapter 1 where he introduces this notation.
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
+1: I spent half an hour to see if Rudin uses $subseteq$ to no avail.
$endgroup$
– Alberto Takase
Feb 3 at 9:05
$begingroup$
@AlbertoTakase some books have an index of notations and many have an introductory chapter like Rudin to settle on notations. Too bad people skip them.
$endgroup$
– Henno Brandsma
Feb 3 at 9:13
$begingroup$
I looked at books and published papers; he was consistent. It seems $subseteq$ is modern, but even recently authors (e.g. Munkres) favor $subset$.
$endgroup$
– Alberto Takase
Feb 3 at 9:20
add a comment |
$begingroup$
$subset$ just means "subset of" in Rudin, not "proper subset of". He does not use $subseteq$. See page 3 in chapter 1 where he introduces this notation.
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
+1: I spent half an hour to see if Rudin uses $subseteq$ to no avail.
$endgroup$
– Alberto Takase
Feb 3 at 9:05
$begingroup$
@AlbertoTakase some books have an index of notations and many have an introductory chapter like Rudin to settle on notations. Too bad people skip them.
$endgroup$
– Henno Brandsma
Feb 3 at 9:13
$begingroup$
I looked at books and published papers; he was consistent. It seems $subseteq$ is modern, but even recently authors (e.g. Munkres) favor $subset$.
$endgroup$
– Alberto Takase
Feb 3 at 9:20
add a comment |
$begingroup$
$subset$ just means "subset of" in Rudin, not "proper subset of". He does not use $subseteq$. See page 3 in chapter 1 where he introduces this notation.
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$subset$ just means "subset of" in Rudin, not "proper subset of". He does not use $subseteq$. See page 3 in chapter 1 where he introduces this notation.
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
answered Feb 3 at 8:43
Henno BrandsmaHenno Brandsma
116k349127
116k349127
$begingroup$
+1: I spent half an hour to see if Rudin uses $subseteq$ to no avail.
$endgroup$
– Alberto Takase
Feb 3 at 9:05
$begingroup$
@AlbertoTakase some books have an index of notations and many have an introductory chapter like Rudin to settle on notations. Too bad people skip them.
$endgroup$
– Henno Brandsma
Feb 3 at 9:13
$begingroup$
I looked at books and published papers; he was consistent. It seems $subseteq$ is modern, but even recently authors (e.g. Munkres) favor $subset$.
$endgroup$
– Alberto Takase
Feb 3 at 9:20
add a comment |
$begingroup$
+1: I spent half an hour to see if Rudin uses $subseteq$ to no avail.
$endgroup$
– Alberto Takase
Feb 3 at 9:05
$begingroup$
@AlbertoTakase some books have an index of notations and many have an introductory chapter like Rudin to settle on notations. Too bad people skip them.
$endgroup$
– Henno Brandsma
Feb 3 at 9:13
$begingroup$
I looked at books and published papers; he was consistent. It seems $subseteq$ is modern, but even recently authors (e.g. Munkres) favor $subset$.
$endgroup$
– Alberto Takase
Feb 3 at 9:20
$begingroup$
+1: I spent half an hour to see if Rudin uses $subseteq$ to no avail.
$endgroup$
– Alberto Takase
Feb 3 at 9:05
$begingroup$
+1: I spent half an hour to see if Rudin uses $subseteq$ to no avail.
$endgroup$
– Alberto Takase
Feb 3 at 9:05
$begingroup$
@AlbertoTakase some books have an index of notations and many have an introductory chapter like Rudin to settle on notations. Too bad people skip them.
$endgroup$
– Henno Brandsma
Feb 3 at 9:13
$begingroup$
@AlbertoTakase some books have an index of notations and many have an introductory chapter like Rudin to settle on notations. Too bad people skip them.
$endgroup$
– Henno Brandsma
Feb 3 at 9:13
$begingroup$
I looked at books and published papers; he was consistent. It seems $subseteq$ is modern, but even recently authors (e.g. Munkres) favor $subset$.
$endgroup$
– Alberto Takase
Feb 3 at 9:20
$begingroup$
I looked at books and published papers; he was consistent. It seems $subseteq$ is modern, but even recently authors (e.g. Munkres) favor $subset$.
$endgroup$
– Alberto Takase
Feb 3 at 9:20
add a comment |
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1
$begingroup$
Rudin uses $subset$ and $subseteq$ interchangeably.
$endgroup$
– Alberto Takase
Feb 3 at 4:26
$begingroup$
@AlbertoTakase that would explain quite a bit, thanks!
$endgroup$
– Sean Lee
Feb 3 at 4:27
1
$begingroup$
@AlbertoTakase he does not use $subseteq$ at all, see page 3 of his book. He even introduces $in$ as a notation there.
$endgroup$
– Henno Brandsma
Feb 3 at 8:47
$begingroup$
"Usually a mathematician..." would be a more accurate replacement of "Rudin"
$endgroup$
– Alberto Takase
Feb 3 at 9:07