Uniqueness of limit of convergent sequence
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I know that limit of convergent sequence is unique for some spaces like metric spaces, Hausdorff spaces, etc. Is there any space the limit of the convergence of sequence is not unique?
-Thanks.
general-topology
edited Nov 24 at 12:59
Paul Frost
8,1041528
asked Jul 1 '14 at 12:43
ruud
748
add a comment |
up vote
3
down vote
favorite
I know that limit of convergent sequence is unique for some spaces like metric spaces, Hausdorff spaces, etc. Is there any space the limit of the convergence of sequence is not unique?
-Thanks.
general-topology
edited Nov 24 at 12:59
Paul Frost
8,1041528
asked Jul 1 '14 at 12:43
ruud
748
See also math.stackexchange.com/q/2943399.
– Paul Frost
Nov 24 at 12:59
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I know that limit of convergent sequence is unique for some spaces like metric spaces, Hausdorff spaces, etc. Is there any space the limit of the convergence of sequence is not unique?
-Thanks.
general-topology
edited Nov 24 at 12:59
Paul Frost
8,1041528
asked Jul 1 '14 at 12:43
ruud
748
I know that limit of convergent sequence is unique for some spaces like metric spaces, Hausdorff spaces, etc. Is there any space the limit of the convergence of sequence is not unique?
-Thanks.
general-topology
general-topology
edited Nov 24 at 12:59
Paul Frost
8,1041528
asked Jul 1 '14 at 12:43
ruud
748
edited Nov 24 at 12:59
Paul Frost
8,1041528
asked Jul 1 '14 at 12:43
ruud
748
edited Nov 24 at 12:59
Paul Frost
8,1041528
edited Nov 24 at 12:59
Paul Frost
8,1041528
edited Nov 24 at 12:59
Paul Frost
8,1041528
8,1041528
asked Jul 1 '14 at 12:43
ruud
748
asked Jul 1 '14 at 12:43
ruud
748
asked Jul 1 '14 at 12:43
ruud
748
748
See also math.stackexchange.com/q/2943399.
– Paul Frost
Nov 24 at 12:59
add a comment |
See also math.stackexchange.com/q/2943399.
– Paul Frost
Nov 24 at 12:59
See also math.stackexchange.com/q/2943399.
– Paul Frost
Nov 24 at 12:59
See also math.stackexchange.com/q/2943399.
– Paul Frost
Nov 24 at 12:59
add a comment |
3 Answers
3
active
oldest
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up vote
3
down vote
accepted
Here is another informative example.
Consider $X=mathbb R$ with the open sets $tau={ mathbb R,emptyset }$. Then a given sequence $(x_n)_{nin mathbb N}$ converges to every real number!
answered Jul 1 '14 at 15:34
Marm
2,83411021
1
More generally, every point in a space equipped with the trivial topology is a limit of every sequence.
– Aleš Bizjak
Jul 1 '14 at 15:44
add a comment |
up vote
3
down vote
Suppose $X$ is infinite and has the cofinite topology. That is, $S subset X$ is closed if and only if $|S|$ is finite or $S=X$. Then, if $s_n$ is a sequence in $X$ with this topology, then $s_nrightarrow s$ for every $s in X$.
So, limits are not unique.
NB: If $X$ is finite, then the cofinite topology is just the discrete topology.
answered Jul 1 '14 at 13:38
MRicci
1,503814
1
The sequence must take on infinite distinct values.
– egreg
Jul 1 '14 at 13:54
add a comment |
up vote
0
down vote
Let X be a non empty set of cardinality greater than 1. Then in indiscrete topological space (X,I) every sequence converges to every point of X and hence limit of convergence sequence is not unique.
answered Nov 24 at 10:14
Sushil Pandit
1
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Here is another informative example.
Consider $X=mathbb R$ with the open sets $tau={ mathbb R,emptyset }$. Then a given sequence $(x_n)_{nin mathbb N}$ converges to every real number!
answered Jul 1 '14 at 15:34
Marm
2,83411021
1
More generally, every point in a space equipped with the trivial topology is a limit of every sequence.
– Aleš Bizjak
Jul 1 '14 at 15:44
add a comment |
up vote
3
down vote
accepted
Here is another informative example.
Consider $X=mathbb R$ with the open sets $tau={ mathbb R,emptyset }$. Then a given sequence $(x_n)_{nin mathbb N}$ converges to every real number!
answered Jul 1 '14 at 15:34
Marm
2,83411021
1
More generally, every point in a space equipped with the trivial topology is a limit of every sequence.
– Aleš Bizjak
Jul 1 '14 at 15:44
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Here is another informative example.
Consider $X=mathbb R$ with the open sets $tau={ mathbb R,emptyset }$. Then a given sequence $(x_n)_{nin mathbb N}$ converges to every real number!
answered Jul 1 '14 at 15:34
Marm
2,83411021
Here is another informative example.
Consider $X=mathbb R$ with the open sets $tau={ mathbb R,emptyset }$. Then a given sequence $(x_n)_{nin mathbb N}$ converges to every real number!
answered Jul 1 '14 at 15:34
Marm
2,83411021
answered Jul 1 '14 at 15:34
Marm
2,83411021
answered Jul 1 '14 at 15:34
Marm
2,83411021
answered Jul 1 '14 at 15:34
Marm
2,83411021
2,83411021
1
More generally, every point in a space equipped with the trivial topology is a limit of every sequence.
– Aleš Bizjak
Jul 1 '14 at 15:44
add a comment |
1
More generally, every point in a space equipped with the trivial topology is a limit of every sequence.
– Aleš Bizjak
Jul 1 '14 at 15:44
1
1
More generally, every point in a space equipped with the trivial topology is a limit of every sequence.
– Aleš Bizjak
Jul 1 '14 at 15:44
More generally, every point in a space equipped with the trivial topology is a limit of every sequence.
– Aleš Bizjak
Jul 1 '14 at 15:44
add a comment |
up vote
3
down vote
Suppose $X$ is infinite and has the cofinite topology. That is, $S subset X$ is closed if and only if $|S|$ is finite or $S=X$. Then, if $s_n$ is a sequence in $X$ with this topology, then $s_nrightarrow s$ for every $s in X$.
So, limits are not unique.
NB: If $X$ is finite, then the cofinite topology is just the discrete topology.
answered Jul 1 '14 at 13:38
MRicci
1,503814
1
The sequence must take on infinite distinct values.
– egreg
Jul 1 '14 at 13:54
add a comment |
up vote
3
down vote
Suppose $X$ is infinite and has the cofinite topology. That is, $S subset X$ is closed if and only if $|S|$ is finite or $S=X$. Then, if $s_n$ is a sequence in $X$ with this topology, then $s_nrightarrow s$ for every $s in X$.
So, limits are not unique.
NB: If $X$ is finite, then the cofinite topology is just the discrete topology.
answered Jul 1 '14 at 13:38
MRicci
1,503814
1
The sequence must take on infinite distinct values.
– egreg
Jul 1 '14 at 13:54
add a comment |
up vote
3
down vote
up vote
3
down vote
Suppose $X$ is infinite and has the cofinite topology. That is, $S subset X$ is closed if and only if $|S|$ is finite or $S=X$. Then, if $s_n$ is a sequence in $X$ with this topology, then $s_nrightarrow s$ for every $s in X$.
So, limits are not unique.
NB: If $X$ is finite, then the cofinite topology is just the discrete topology.
answered Jul 1 '14 at 13:38
MRicci
1,503814
Suppose $X$ is infinite and has the cofinite topology. That is, $S subset X$ is closed if and only if $|S|$ is finite or $S=X$. Then, if $s_n$ is a sequence in $X$ with this topology, then $s_nrightarrow s$ for every $s in X$.
So, limits are not unique.
NB: If $X$ is finite, then the cofinite topology is just the discrete topology.
answered Jul 1 '14 at 13:38
MRicci
1,503814
edited Jul 1 '14 at 13:45
answered Jul 1 '14 at 13:38
MRicci
1,503814
answered Jul 1 '14 at 13:38
MRicci
1,503814
answered Jul 1 '14 at 13:38
MRicci
1,503814
1,503814
1
The sequence must take on infinite distinct values.
– egreg
Jul 1 '14 at 13:54
add a comment |
1
The sequence must take on infinite distinct values.
– egreg
Jul 1 '14 at 13:54
1
1
The sequence must take on infinite distinct values.
– egreg
Jul 1 '14 at 13:54
The sequence must take on infinite distinct values.
– egreg
Jul 1 '14 at 13:54
add a comment |
up vote
0
down vote
Let X be a non empty set of cardinality greater than 1. Then in indiscrete topological space (X,I) every sequence converges to every point of X and hence limit of convergence sequence is not unique.
answered Nov 24 at 10:14
Sushil Pandit
1
add a comment |
up vote
0
down vote
Let X be a non empty set of cardinality greater than 1. Then in indiscrete topological space (X,I) every sequence converges to every point of X and hence limit of convergence sequence is not unique.
answered Nov 24 at 10:14
Sushil Pandit
1
add a comment |
up vote
0
down vote
up vote
0
down vote
Let X be a non empty set of cardinality greater than 1. Then in indiscrete topological space (X,I) every sequence converges to every point of X and hence limit of convergence sequence is not unique.
answered Nov 24 at 10:14
Sushil Pandit
1
Let X be a non empty set of cardinality greater than 1. Then in indiscrete topological space (X,I) every sequence converges to every point of X and hence limit of convergence sequence is not unique.
answered Nov 24 at 10:14
Sushil Pandit
1
answered Nov 24 at 10:14
Sushil Pandit
1
answered Nov 24 at 10:14
Sushil Pandit
1
answered Nov 24 at 10:14
Sushil Pandit
1
1
add a comment |
add a comment |
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See also math.stackexchange.com/q/2943399.
– Paul Frost
Nov 24 at 12:59