Mixing
In topological proofs, are any subsets mentioned always subsets comprising the topology?
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I am trying to learn, through self-study, about topological manifolds for the first time and would appreciate someone clarifying a point for me.
In proofs about topological manifolds there are frequent references made to subsets. Are the referenced subsets always one or more of the subsets that comprise the topology, or can they be special subsets that carve out new spaces in the topology but are not actual subsets of the topology?
For example, are balls, or subsets that are dense in a topology, always among those that constitute the topology?
general-topology
asked Jan 28 at 20:20
MarcellusMarcellus
11
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add a comment |
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I am trying to learn, through self-study, about topological manifolds for the first time and would appreciate someone clarifying a point for me.
In proofs about topological manifolds there are frequent references made to subsets. Are the referenced subsets always one or more of the subsets that comprise the topology, or can they be special subsets that carve out new spaces in the topology but are not actual subsets of the topology?
For example, are balls, or subsets that are dense in a topology, always among those that constitute the topology?
general-topology
asked Jan 28 at 20:20
MarcellusMarcellus
11
$endgroup$
$begingroup$
If we talk about the standard topology on the real numbers, clearly $mathbb{Z} subset mathbb{R}$ but are the integers an open set (an element of the topology)?
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– Good Morning Captain
Jan 28 at 20:32
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No. If we want to refer to such a subset, we call it open subset.
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– Hagen von Eitzen
Jan 28 at 20:41
add a comment |
$begingroup$
I am trying to learn, through self-study, about topological manifolds for the first time and would appreciate someone clarifying a point for me.
In proofs about topological manifolds there are frequent references made to subsets. Are the referenced subsets always one or more of the subsets that comprise the topology, or can they be special subsets that carve out new spaces in the topology but are not actual subsets of the topology?
For example, are balls, or subsets that are dense in a topology, always among those that constitute the topology?
general-topology
asked Jan 28 at 20:20
MarcellusMarcellus
11
$endgroup$
I am trying to learn, through self-study, about topological manifolds for the first time and would appreciate someone clarifying a point for me.
In proofs about topological manifolds there are frequent references made to subsets. Are the referenced subsets always one or more of the subsets that comprise the topology, or can they be special subsets that carve out new spaces in the topology but are not actual subsets of the topology?
For example, are balls, or subsets that are dense in a topology, always among those that constitute the topology?
general-topology
general-topology
asked Jan 28 at 20:20
MarcellusMarcellus
11
asked Jan 28 at 20:20
MarcellusMarcellus
11
asked Jan 28 at 20:20
MarcellusMarcellus
11
asked Jan 28 at 20:20
MarcellusMarcellus
11
asked Jan 28 at 20:20
MarcellusMarcellus
11
11
$begingroup$
If we talk about the standard topology on the real numbers, clearly $mathbb{Z} subset mathbb{R}$ but are the integers an open set (an element of the topology)?
$endgroup$
– Good Morning Captain
Jan 28 at 20:32
$begingroup$
No. If we want to refer to such a subset, we call it open subset.
$endgroup$
– Hagen von Eitzen
Jan 28 at 20:41
add a comment |
$begingroup$
If we talk about the standard topology on the real numbers, clearly $mathbb{Z} subset mathbb{R}$ but are the integers an open set (an element of the topology)?
$endgroup$
– Good Morning Captain
Jan 28 at 20:32
$begingroup$
No. If we want to refer to such a subset, we call it open subset.
$endgroup$
– Hagen von Eitzen
Jan 28 at 20:41
$begingroup$
If we talk about the standard topology on the real numbers, clearly $mathbb{Z} subset mathbb{R}$ but are the integers an open set (an element of the topology)?
$endgroup$
– Good Morning Captain
Jan 28 at 20:32
$begingroup$
If we talk about the standard topology on the real numbers, clearly $mathbb{Z} subset mathbb{R}$ but are the integers an open set (an element of the topology)?
$endgroup$
– Good Morning Captain
Jan 28 at 20:32
$begingroup$
No. If we want to refer to such a subset, we call it open subset.
$endgroup$
– Hagen von Eitzen
Jan 28 at 20:41
$begingroup$
No. If we want to refer to such a subset, we call it open subset.
$endgroup$
– Hagen von Eitzen
Jan 28 at 20:41
add a comment |
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$begingroup$
If we talk about the standard topology on the real numbers, clearly $mathbb{Z} subset mathbb{R}$ but are the integers an open set (an element of the topology)?
$endgroup$
– Good Morning Captain
Jan 28 at 20:32
$begingroup$
No. If we want to refer to such a subset, we call it open subset.
$endgroup$
– Hagen von Eitzen
Jan 28 at 20:41