Why is $emptysetcupnmathbbZ, nin mathbbN$ not a topology on $mathbbZ$?Is it true that, for accumulation points, $(A cup B)' subset A' cup B'$?Testing for topology on $mathbb R^2$Verifying that a certain collection of intervals of $mathbb R$ forms a topologyProving that $[-n, n] mid n in mathbb N cup varnothing, mathbb R$ is a topology over $mathbb R$Why axiom 3) of topology is redundant?Show that $tau= emptyset, mathbbR cup (-infty,x]:xinmathbbR$ is not a topology on $mathbbR$Homeomorphism of $K$ and $Kcup 0$Show that $tau(frac1n)$ is a topology for $mathbbR$.Proof that the Michael line is a topology.Let $X=BbbN setminus lbrace 1 rbrace$, $A_n=lbrace din X : d|n rbrace$, for $nin BbbN$. Is $tau=lbrace A_n : nin BbbN$ topology?
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Why is $emptysetcupnmathbbZ, nin mathbbN$ not a topology on $mathbbZ$?
Is it true that, for accumulation points, $(A cup B)' subset A' cup B'$?Testing for topology on $mathbb R^2$Verifying that a certain collection of intervals of $mathbb R$ forms a topologyProving that $[-n, n] mid n in mathbb N cup varnothing, mathbb R$ is a topology over $mathbb R$Why axiom 3) of topology is redundant?Show that $tau= emptyset, mathbbR cup (-infty,x]:xinmathbbR$ is not a topology on $mathbbR$Homeomorphism of $K$ and $Kcup 0$Show that $tau(frac1n)$ is a topology for $mathbbR$.Proof that the Michael line is a topology.Let $X=BbbN setminus lbrace 1 rbrace$, $A_n=lbrace din X : d|n rbrace$, for $nin BbbN$. Is $tau=lbrace A_n : nin BbbN$ topology?
$begingroup$
please why $tau=emptysetcupnmathbbZ, nin mathbbN$ is not a topology on $mathbbZ$?
$emptyset in mathbbZ$ and $mathbbZ=1mathbbZin tau$
let $A=n_1mathbbZ$ and $B=n_2mathbbZ$ then $Acap B= maxn_1,n_2mathbbZin tau$
then $tau$ is stable by finite intersection .
let $(A_i)_iin I$ a family of sets from $tau$ : $A_i=n_imathbbZ$
then $bigcup A_i=min_Iin In_i mathbbZ$
thank you
general-topology
$endgroup$
add a comment |
$begingroup$
please why $tau=emptysetcupnmathbbZ, nin mathbbN$ is not a topology on $mathbbZ$?
$emptyset in mathbbZ$ and $mathbbZ=1mathbbZin tau$
let $A=n_1mathbbZ$ and $B=n_2mathbbZ$ then $Acap B= maxn_1,n_2mathbbZin tau$
then $tau$ is stable by finite intersection .
let $(A_i)_iin I$ a family of sets from $tau$ : $A_i=n_imathbbZ$
then $bigcup A_i=min_Iin In_i mathbbZ$
thank you
general-topology
$endgroup$
add a comment |
$begingroup$
please why $tau=emptysetcupnmathbbZ, nin mathbbN$ is not a topology on $mathbbZ$?
$emptyset in mathbbZ$ and $mathbbZ=1mathbbZin tau$
let $A=n_1mathbbZ$ and $B=n_2mathbbZ$ then $Acap B= maxn_1,n_2mathbbZin tau$
then $tau$ is stable by finite intersection .
let $(A_i)_iin I$ a family of sets from $tau$ : $A_i=n_imathbbZ$
then $bigcup A_i=min_Iin In_i mathbbZ$
thank you
general-topology
$endgroup$
please why $tau=emptysetcupnmathbbZ, nin mathbbN$ is not a topology on $mathbbZ$?
$emptyset in mathbbZ$ and $mathbbZ=1mathbbZin tau$
let $A=n_1mathbbZ$ and $B=n_2mathbbZ$ then $Acap B= maxn_1,n_2mathbbZin tau$
then $tau$ is stable by finite intersection .
let $(A_i)_iin I$ a family of sets from $tau$ : $A_i=n_imathbbZ$
then $bigcup A_i=min_Iin In_i mathbbZ$
thank you
general-topology
general-topology
edited Mar 11 at 20:49
Asaf Karagila♦
306k33438769
306k33438769
asked Mar 11 at 20:43
Poline SandraPoline Sandra
997
997
add a comment |
add a comment |
3 Answers
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$begingroup$
$2mathbbZcup 3mathbbZne nmathbbZ$ for any $n$.
$endgroup$
$begingroup$
why ? I don't understand ? and the intersection is right? ?
$endgroup$
– Poline Sandra
Mar 11 at 20:52
$begingroup$
@PolineSandra your family is a base for a topology but not a topology. As to this non-equality: $2$ is in the left hand side and can only be in $nmathbbZ$ for $n=1,2$ but the first is too large (as $5$ is not in the left hand side), the second too small as $3$ is in the left hand side and not in $2mathbbZ$.
$endgroup$
– Henno Brandsma
Mar 11 at 22:40
add a comment |
$begingroup$
It is not true that $cup _i in IA_i = min_i in In_i mathbbZ$. Can you come up with a counter-example where $I$ has size $2$?
New contributor
$endgroup$
add a comment |
$begingroup$
As you said, $emptyset in tau$ and $mathbbZ=1mathbbZ in tau$.
Also, let $A=amathbbZ$ and $B=bmathbbZ$. We have that $A cap B = textrmlcm(a,b)mathbbZ in tau$.
But this set is not a topology, because it doesn't satisfy the union property. Just take any $A=amathbbZ$ and $B=bmathbbZ$ such that $textrmgcd(a,b)=1$. For example: $3mathbbZ cup 5mathbbZ neq nmathbbZ, forall n in mathbbZ$. Note that this union contains all integers that are multiple to $3$ and all integers that are multiple to $5$.
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add a comment |
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3 Answers
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3 Answers
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$begingroup$
$2mathbbZcup 3mathbbZne nmathbbZ$ for any $n$.
$endgroup$
$begingroup$
why ? I don't understand ? and the intersection is right? ?
$endgroup$
– Poline Sandra
Mar 11 at 20:52
$begingroup$
@PolineSandra your family is a base for a topology but not a topology. As to this non-equality: $2$ is in the left hand side and can only be in $nmathbbZ$ for $n=1,2$ but the first is too large (as $5$ is not in the left hand side), the second too small as $3$ is in the left hand side and not in $2mathbbZ$.
$endgroup$
– Henno Brandsma
Mar 11 at 22:40
add a comment |
$begingroup$
$2mathbbZcup 3mathbbZne nmathbbZ$ for any $n$.
$endgroup$
$begingroup$
why ? I don't understand ? and the intersection is right? ?
$endgroup$
– Poline Sandra
Mar 11 at 20:52
$begingroup$
@PolineSandra your family is a base for a topology but not a topology. As to this non-equality: $2$ is in the left hand side and can only be in $nmathbbZ$ for $n=1,2$ but the first is too large (as $5$ is not in the left hand side), the second too small as $3$ is in the left hand side and not in $2mathbbZ$.
$endgroup$
– Henno Brandsma
Mar 11 at 22:40
add a comment |
$begingroup$
$2mathbbZcup 3mathbbZne nmathbbZ$ for any $n$.
$endgroup$
$2mathbbZcup 3mathbbZne nmathbbZ$ for any $n$.
answered Mar 11 at 20:46
eyeballfrogeyeballfrog
6,471629
6,471629
$begingroup$
why ? I don't understand ? and the intersection is right? ?
$endgroup$
– Poline Sandra
Mar 11 at 20:52
$begingroup$
@PolineSandra your family is a base for a topology but not a topology. As to this non-equality: $2$ is in the left hand side and can only be in $nmathbbZ$ for $n=1,2$ but the first is too large (as $5$ is not in the left hand side), the second too small as $3$ is in the left hand side and not in $2mathbbZ$.
$endgroup$
– Henno Brandsma
Mar 11 at 22:40
add a comment |
$begingroup$
why ? I don't understand ? and the intersection is right? ?
$endgroup$
– Poline Sandra
Mar 11 at 20:52
$begingroup$
@PolineSandra your family is a base for a topology but not a topology. As to this non-equality: $2$ is in the left hand side and can only be in $nmathbbZ$ for $n=1,2$ but the first is too large (as $5$ is not in the left hand side), the second too small as $3$ is in the left hand side and not in $2mathbbZ$.
$endgroup$
– Henno Brandsma
Mar 11 at 22:40
$begingroup$
why ? I don't understand ? and the intersection is right? ?
$endgroup$
– Poline Sandra
Mar 11 at 20:52
$begingroup$
why ? I don't understand ? and the intersection is right? ?
$endgroup$
– Poline Sandra
Mar 11 at 20:52
$begingroup$
@PolineSandra your family is a base for a topology but not a topology. As to this non-equality: $2$ is in the left hand side and can only be in $nmathbbZ$ for $n=1,2$ but the first is too large (as $5$ is not in the left hand side), the second too small as $3$ is in the left hand side and not in $2mathbbZ$.
$endgroup$
– Henno Brandsma
Mar 11 at 22:40
$begingroup$
@PolineSandra your family is a base for a topology but not a topology. As to this non-equality: $2$ is in the left hand side and can only be in $nmathbbZ$ for $n=1,2$ but the first is too large (as $5$ is not in the left hand side), the second too small as $3$ is in the left hand side and not in $2mathbbZ$.
$endgroup$
– Henno Brandsma
Mar 11 at 22:40
add a comment |
$begingroup$
It is not true that $cup _i in IA_i = min_i in In_i mathbbZ$. Can you come up with a counter-example where $I$ has size $2$?
New contributor
$endgroup$
add a comment |
$begingroup$
It is not true that $cup _i in IA_i = min_i in In_i mathbbZ$. Can you come up with a counter-example where $I$ has size $2$?
New contributor
$endgroup$
add a comment |
$begingroup$
It is not true that $cup _i in IA_i = min_i in In_i mathbbZ$. Can you come up with a counter-example where $I$ has size $2$?
New contributor
$endgroup$
It is not true that $cup _i in IA_i = min_i in In_i mathbbZ$. Can you come up with a counter-example where $I$ has size $2$?
New contributor
New contributor
answered Mar 11 at 20:46
nammienammie
2399
2399
New contributor
New contributor
add a comment |
add a comment |
$begingroup$
As you said, $emptyset in tau$ and $mathbbZ=1mathbbZ in tau$.
Also, let $A=amathbbZ$ and $B=bmathbbZ$. We have that $A cap B = textrmlcm(a,b)mathbbZ in tau$.
But this set is not a topology, because it doesn't satisfy the union property. Just take any $A=amathbbZ$ and $B=bmathbbZ$ such that $textrmgcd(a,b)=1$. For example: $3mathbbZ cup 5mathbbZ neq nmathbbZ, forall n in mathbbZ$. Note that this union contains all integers that are multiple to $3$ and all integers that are multiple to $5$.
$endgroup$
add a comment |
$begingroup$
As you said, $emptyset in tau$ and $mathbbZ=1mathbbZ in tau$.
Also, let $A=amathbbZ$ and $B=bmathbbZ$. We have that $A cap B = textrmlcm(a,b)mathbbZ in tau$.
But this set is not a topology, because it doesn't satisfy the union property. Just take any $A=amathbbZ$ and $B=bmathbbZ$ such that $textrmgcd(a,b)=1$. For example: $3mathbbZ cup 5mathbbZ neq nmathbbZ, forall n in mathbbZ$. Note that this union contains all integers that are multiple to $3$ and all integers that are multiple to $5$.
$endgroup$
add a comment |
$begingroup$
As you said, $emptyset in tau$ and $mathbbZ=1mathbbZ in tau$.
Also, let $A=amathbbZ$ and $B=bmathbbZ$. We have that $A cap B = textrmlcm(a,b)mathbbZ in tau$.
But this set is not a topology, because it doesn't satisfy the union property. Just take any $A=amathbbZ$ and $B=bmathbbZ$ such that $textrmgcd(a,b)=1$. For example: $3mathbbZ cup 5mathbbZ neq nmathbbZ, forall n in mathbbZ$. Note that this union contains all integers that are multiple to $3$ and all integers that are multiple to $5$.
$endgroup$
As you said, $emptyset in tau$ and $mathbbZ=1mathbbZ in tau$.
Also, let $A=amathbbZ$ and $B=bmathbbZ$. We have that $A cap B = textrmlcm(a,b)mathbbZ in tau$.
But this set is not a topology, because it doesn't satisfy the union property. Just take any $A=amathbbZ$ and $B=bmathbbZ$ such that $textrmgcd(a,b)=1$. For example: $3mathbbZ cup 5mathbbZ neq nmathbbZ, forall n in mathbbZ$. Note that this union contains all integers that are multiple to $3$ and all integers that are multiple to $5$.
answered Mar 11 at 21:05
White CrowWhite Crow
284110
284110
add a comment |
add a comment |
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