First thank you in advance. I am reading Munkres book on Topology pg 96 example 6 gives some examples of sets which are closed. I do not fully understand why these examples are closed, and am looking for a bit of clarification.

definition : The closure of a set is defined as all the intersections of all closed sets containing A. (so this closure set, is a family of sets. * correction, the closure is the smallest set in the intersection of the family of sets containing A*)

Remember that a set is closed , A, if Y -A is open , for (Y,$\tau$) a topological space on Y.

  1. B = $\{\frac{1}{n} | n \in \mathbb{Z}+\}$ where $\bar{B} = \{0\} \bigcup B$
  2. C = $\{0\} \bigcup (1,2)$ and $\bar{C} = \{0\} \bigcup [1,2]$.

My question boils down for example 1, what are the closed sets on B? If i can find that, then I can intersect them. And I am not sure how they derived 2.

Thank you again for any hints.

  • $\begingroup$ "so this closure set, is a family of sets". No: it is the intersection of a family of sets, or simply put, the smallest set of this collection. $\endgroup$
    – Pedro Tamaroff
    Sep 12 '14 at 2:06
  • 1
    $\begingroup$ @PedroTamaroff the smallest set in that collection need not a-priori exist. $\endgroup$ Sep 12 '14 at 2:07
  • 1
    $\begingroup$ @AnthonyColombo I think you should stop for a minute and figure out why in the Euclidean topology on $\mathbb R$ the closure of $(a,b)$ is $[a,b]$. It will sort things out for you. $\endgroup$ Sep 12 '14 at 2:08
  • $\begingroup$ and while I didn't check I doubt there is a topology book written by Munkres and by Muncres. $\endgroup$ Sep 12 '14 at 2:11
  • $\begingroup$ every detail is important as the idea. thank you $\endgroup$ Sep 12 '14 at 2:12

okay, the way i understand 2. is that first consider (a,b) as an open interval, then the intersection of all closed sets [a,b] contains (a,b)... however the closure must contain the set C. thus if one performs the union of the element {0} you can obtain the closure C given as ${0} \bigcup [a,b]$ which for all intersections will contain C.


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