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?










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    0












    $begingroup$


    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?










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      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?










      share|cite|improve this question









      $endgroup$




      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






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      asked Dec 18 '18 at 5:51









      BDSubBDSub

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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.






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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.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              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.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                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.






                share|cite|improve this answer









                $endgroup$



                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.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 18 '18 at 6:03









                Robert ThingumRobert Thingum

                9561317




                9561317






























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