-1

I'm confused b/w the structure of standard topology on $\mathbb R $ i.e,$\mathbb R $ & the structure of K-topology on $\mathbb R $ i.e,$\mathbb R_K $

Since, standard topology on $\mathbb R$ is generated by (a,b),whereas the K-topology on $\mathbb R$ $(a,b)\cup(a,b)-$K.

But,$(a,b)\cup(a,b)-$K=$(a,b)$.

Therefore,standard topology on $ \mathbb R$ and $\mathbb R_K$ are generated by the collection of open intervals on the real line.Hence,Standard topology on $\mathbb R$=$\mathbb R_K$.

But,this result is wrong,as $\mathbb R_K$ is strictly finer than the Standard topology on $\mathbb R$ .

NOTE

$K-$topology on $\mathbb R$==let $K$ denote the set of all numbers of the form $1/n$,for $n \in \mathbb Z_+$,and let β′′ be the collection of all open intervals (a,b),along with all sets of the form (a,b)−K.The topology generated by β′′ will be called K−topology on R

Standard topology on $\mathbb R$==let β be the collection of open intervals in the real line,$(a,b)={x:a<x<b}$, the topology generated by β is called the standard topology on $\mathbb R$.

Help me in clearing this concept,i've just started studying topology.

PVAL-inactive
  • 8,551
Styles
  • 3,539
  • 1
    What is the $K$- topology? – Aweygan Aug 08 '16 at 19:38
  • @Aweygan:let $K$ denote the set of all numbers of the form $1/n$,for $n \in \mathbb Z_+$,and let $\beta'' $ be the collection of all open intervals $(a,b)$,along with all sets of the form $(a,b)-K$,The topology generated by $\beta''$ will be called $K-$topology on $\mathbb R$. – Styles Aug 08 '16 at 19:48
  • 1
    Don't you think that would have been important to mention in the body of your question? – Aweygan Aug 08 '16 at 19:52
  • @Aweygan:I thought that these are the standard terminologies.But,anyways i'll mention it & sorry for inconvenience( :->). – Styles Aug 08 '16 at 19:55
  • It's fine, but the notation $K={1/n:n\in\mathbb{N}}$ is not standard, nor (as far as I'm aware) is the notation $K$-topology, for an arbitrary subset $K$. – Aweygan Aug 08 '16 at 20:01
  • 1
    Possible duplicate of http://math.stackexchange.com/questions/279027/are-there-sets-in-the-k-topology-that-arent-open-in-the-standard-topology –  Aug 08 '16 at 21:04

2 Answers2

1

Take $U=(-1,1)\setminus K$ open interval in $\mathbb{R_k}$ but it is open in $\mathbb{R}$.

amWhy
  • 209,954
Gob
  • 1,117
  • 1
    :thanks for your response but your answer can help in showing $\mathbb R_K$ is strictly finer than the standard toplogy on $\mathbb R$ (with which i agree).My problem is if $\mathbb R_K$ & standard toplogy on $\mathbb R$ are generated by same objects,then why they are considered different? – Styles Aug 08 '16 at 20:26
  • They are not generated by the same objects the biases are different. My advice see Munkers page 82. We will see the argument. – Gob Aug 08 '16 at 22:03
  • :it will be better if you edit your answer. – Styles Aug 08 '16 at 22:17
1

@amWhy & Battani:How can $(-1,1)\setminus K$ is open in both $\mathbb{R_K}$ & usual topology in $\mathbb{R}$?