Is there a name for this type of topology?
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Let $tau$ be a topology on $X$ such that for every $a, b$ in $X$ there exists a bijective function $f$ that is continuous and has a continuous inverse such that $f(a) = b$.
Examples of such a topology:
the topology induced by the euclidean metric on $mathbb R^n$
the discrete topology, the trivial topology
non examples:
$tau = emptyset,a,b,c,X$ where $X = a,b,c$
Is there a name for these type of topologies?
general-topology
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
asked Dec 28 '17 at 20:41
mathew
400115
add a comment |Â
up vote
4
down vote
favorite
Let $tau$ be a topology on $X$ such that for every $a, b$ in $X$ there exists a bijective function $f$ that is continuous and has a continuous inverse such that $f(a) = b$.
Examples of such a topology:
the topology induced by the euclidean metric on $mathbb R^n$
the discrete topology, the trivial topology
non examples:
$tau = emptyset,a,b,c,X$ where $X = a,b,c$
Is there a name for these type of topologies?
general-topology
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
asked Dec 28 '17 at 20:41
mathew
400115
3
It seems that such spaces are called homogeneous.
– Sangchul Lee
Dec 28 '17 at 20:56
yes that appears to be what I want, thank you, feel free to write that as an answer to the question
– mathew
Dec 28 '17 at 21:00
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Let $tau$ be a topology on $X$ such that for every $a, b$ in $X$ there exists a bijective function $f$ that is continuous and has a continuous inverse such that $f(a) = b$.
Examples of such a topology:
the topology induced by the euclidean metric on $mathbb R^n$
the discrete topology, the trivial topology
non examples:
$tau = emptyset,a,b,c,X$ where $X = a,b,c$
Is there a name for these type of topologies?
general-topology
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
asked Dec 28 '17 at 20:41
mathew
400115
Let $tau$ be a topology on $X$ such that for every $a, b$ in $X$ there exists a bijective function $f$ that is continuous and has a continuous inverse such that $f(a) = b$.
Examples of such a topology:
the topology induced by the euclidean metric on $mathbb R^n$
the discrete topology, the trivial topology
non examples:
$tau = emptyset,a,b,c,X$ where $X = a,b,c$
Is there a name for these type of topologies?
general-topology
general-topology
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
asked Dec 28 '17 at 20:41
mathew
400115
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
asked Dec 28 '17 at 20:41
mathew
400115
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
edited Dec 30 '17 at 0:24
Theoretical Economist
3,6032730
3,6032730
asked Dec 28 '17 at 20:41
mathew
400115
asked Dec 28 '17 at 20:41
mathew
400115
asked Dec 28 '17 at 20:41
mathew
400115
400115
3
It seems that such spaces are called homogeneous.
– Sangchul Lee
Dec 28 '17 at 20:56
yes that appears to be what I want, thank you, feel free to write that as an answer to the question
– mathew
Dec 28 '17 at 21:00
add a comment |Â
3
It seems that such spaces are called homogeneous.
– Sangchul Lee
Dec 28 '17 at 20:56
yes that appears to be what I want, thank you, feel free to write that as an answer to the question
– mathew
Dec 28 '17 at 21:00
3
3
It seems that such spaces are called homogeneous.
– Sangchul Lee
Dec 28 '17 at 20:56
It seems that such spaces are called homogeneous.
– Sangchul Lee
Dec 28 '17 at 20:56
yes that appears to be what I want, thank you, feel free to write that as an answer to the question
– mathew
Dec 28 '17 at 21:00
yes that appears to be what I want, thank you, feel free to write that as an answer to the question
– mathew
Dec 28 '17 at 21:00
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
6
down vote
accepted
As Sangchul says in the comments, such spaces are called homogeneous. An equivalent way to phrase the definition is that the group of homeomorphisms $X to X$ acts transitively on $X$. Important examples include
- any connected manifold (this is not obvious)
- any topological group $G$
- any quotient $G/H$ of a topological group by a subgroup.
The term "homogeneous space" is also used for something more specific, namely a pair consisting of a space $X$ and a topological (sometimes Lie) group $G$ acting transitively on it.
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
As Sangchul says in the comments, such spaces are called homogeneous. An equivalent way to phrase the definition is that the group of homeomorphisms $X to X$ acts transitively on $X$. Important examples include
- any connected manifold (this is not obvious)
- any topological group $G$
- any quotient $G/H$ of a topological group by a subgroup.
The term "homogeneous space" is also used for something more specific, namely a pair consisting of a space $X$ and a topological (sometimes Lie) group $G$ acting transitively on it.
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
add a comment |Â
up vote
6
down vote
accepted
As Sangchul says in the comments, such spaces are called homogeneous. An equivalent way to phrase the definition is that the group of homeomorphisms $X to X$ acts transitively on $X$. Important examples include
- any connected manifold (this is not obvious)
- any topological group $G$
- any quotient $G/H$ of a topological group by a subgroup.
The term "homogeneous space" is also used for something more specific, namely a pair consisting of a space $X$ and a topological (sometimes Lie) group $G$ acting transitively on it.
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
add a comment |Â
up vote
6
down vote
accepted
up vote
6
down vote
accepted
As Sangchul says in the comments, such spaces are called homogeneous. An equivalent way to phrase the definition is that the group of homeomorphisms $X to X$ acts transitively on $X$. Important examples include
- any connected manifold (this is not obvious)
- any topological group $G$
- any quotient $G/H$ of a topological group by a subgroup.
The term "homogeneous space" is also used for something more specific, namely a pair consisting of a space $X$ and a topological (sometimes Lie) group $G$ acting transitively on it.
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
As Sangchul says in the comments, such spaces are called homogeneous. An equivalent way to phrase the definition is that the group of homeomorphisms $X to X$ acts transitively on $X$. Important examples include
- any connected manifold (this is not obvious)
- any topological group $G$
- any quotient $G/H$ of a topological group by a subgroup.
The term "homogeneous space" is also used for something more specific, namely a pair consisting of a space $X$ and a topological (sometimes Lie) group $G$ acting transitively on it.
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
edited Sep 8 at 5:29
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
answered Dec 28 '17 at 23:28
Qiaochu Yuan
270k32567904
270k32567904
add a comment |Â
add a comment |Â
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3
It seems that such spaces are called homogeneous.
– Sangchul Lee
Dec 28 '17 at 20:56
yes that appears to be what I want, thank you, feel free to write that as an answer to the question
– mathew
Dec 28 '17 at 21:00