Definition and Notion for Basis of Topology
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I try to figure out the equivalence between the definitions of basis for topology given in Munkres 2nd edition and wikipedia.
In wikipedia;
A base is a collection $mathcal{B}$ of subsets of $X$ satisfying the following properties:
The base elements cover $X$.
Let $B_1$, $B_2$ be base elements and let $I$ be their intersection. Then for each $x$ in $I$, there is a base element $B_3$ containing $x$ and contained in $I$.
Now, If I take $X$ = { a, b, c } and $mathcal{B}$ = { {a, b, c} } then , is this a basis for any topology?
Since,
i) Element of $mathcal{B}$ covers $X$.
ii) There are not such $B_1$ and $B_2$ as in definition of basis !
My question is; Does this example satisfy the definition of basis?
If yes, Is it necessary to mention the associated topology with basis?
general-topology
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
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add a comment |
$begingroup$
I try to figure out the equivalence between the definitions of basis for topology given in Munkres 2nd edition and wikipedia.
In wikipedia;
A base is a collection $mathcal{B}$ of subsets of $X$ satisfying the following properties:
The base elements cover $X$.
Let $B_1$, $B_2$ be base elements and let $I$ be their intersection. Then for each $x$ in $I$, there is a base element $B_3$ containing $x$ and contained in $I$.
Now, If I take $X$ = { a, b, c } and $mathcal{B}$ = { {a, b, c} } then , is this a basis for any topology?
Since,
i) Element of $mathcal{B}$ covers $X$.
ii) There are not such $B_1$ and $B_2$ as in definition of basis !
My question is; Does this example satisfy the definition of basis?
If yes, Is it necessary to mention the associated topology with basis?
general-topology
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
$endgroup$
add a comment |
$begingroup$
I try to figure out the equivalence between the definitions of basis for topology given in Munkres 2nd edition and wikipedia.
In wikipedia;
A base is a collection $mathcal{B}$ of subsets of $X$ satisfying the following properties:
The base elements cover $X$.
Let $B_1$, $B_2$ be base elements and let $I$ be their intersection. Then for each $x$ in $I$, there is a base element $B_3$ containing $x$ and contained in $I$.
Now, If I take $X$ = { a, b, c } and $mathcal{B}$ = { {a, b, c} } then , is this a basis for any topology?
Since,
i) Element of $mathcal{B}$ covers $X$.
ii) There are not such $B_1$ and $B_2$ as in definition of basis !
My question is; Does this example satisfy the definition of basis?
If yes, Is it necessary to mention the associated topology with basis?
general-topology
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
$endgroup$
I try to figure out the equivalence between the definitions of basis for topology given in Munkres 2nd edition and wikipedia.
In wikipedia;
A base is a collection $mathcal{B}$ of subsets of $X$ satisfying the following properties:
The base elements cover $X$.
Let $B_1$, $B_2$ be base elements and let $I$ be their intersection. Then for each $x$ in $I$, there is a base element $B_3$ containing $x$ and contained in $I$.
Now, If I take $X$ = { a, b, c } and $mathcal{B}$ = { {a, b, c} } then , is this a basis for any topology?
Since,
i) Element of $mathcal{B}$ covers $X$.
ii) There are not such $B_1$ and $B_2$ as in definition of basis !
My question is; Does this example satisfy the definition of basis?
If yes, Is it necessary to mention the associated topology with basis?
general-topology
general-topology
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
asked Dec 18 '18 at 5:51
BDSubBDSub
1147
1147
add a comment |
add a comment |
1 Answer
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Yes, it is a basis for precisely the reason you stated. Namely, it satisfies the definition of a basis.
Regarding whether or not you need to state whatever topology, that would be up to personal taste in my opinion. However in many cases you describe a topology in terms of a basis for that topology such as saying that the euclidean topology of the plane is the topology generated by the basis of open balls.
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
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1 Answer
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1 Answer
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$begingroup$
Yes, it is a basis for precisely the reason you stated. Namely, it satisfies the definition of a basis.
Regarding whether or not you need to state whatever topology, that would be up to personal taste in my opinion. However in many cases you describe a topology in terms of a basis for that topology such as saying that the euclidean topology of the plane is the topology generated by the basis of open balls.
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
$endgroup$
add a comment |
$begingroup$
Yes, it is a basis for precisely the reason you stated. Namely, it satisfies the definition of a basis.
Regarding whether or not you need to state whatever topology, that would be up to personal taste in my opinion. However in many cases you describe a topology in terms of a basis for that topology such as saying that the euclidean topology of the plane is the topology generated by the basis of open balls.
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
$endgroup$
add a comment |
$begingroup$
Yes, it is a basis for precisely the reason you stated. Namely, it satisfies the definition of a basis.
Regarding whether or not you need to state whatever topology, that would be up to personal taste in my opinion. However in many cases you describe a topology in terms of a basis for that topology such as saying that the euclidean topology of the plane is the topology generated by the basis of open balls.
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
$endgroup$
Yes, it is a basis for precisely the reason you stated. Namely, it satisfies the definition of a basis.
Regarding whether or not you need to state whatever topology, that would be up to personal taste in my opinion. However in many cases you describe a topology in terms of a basis for that topology such as saying that the euclidean topology of the plane is the topology generated by the basis of open balls.
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
answered Dec 18 '18 at 6:03
Robert ThingumRobert Thingum
7981316
7981316
add a comment |
add a comment |
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