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Describing closures and interiors of subsets of Moore Plane

  1. Aug 13, 2014 #1
    The problem statement, all variables and given/known data.

    Let ##X=\{(x,y) \in \mathbb R^2 :y \geq 0\}##. If ##p=(x,y)## with ##y>0##, let

    ##\mathcal F_p=\{B_r(p) : 0<r<y\}##, and if ##p=(x,0)##, let ##\mathcal F_p=\{B_r(x,r) \cup \{p\}: 0<r\}##.

    Then, there is a neighbourhood filter system generated on ##X## and if ##\tau=\{A \in \mathcal P(X): \forall x\in A, A \in F_p\} \cup \{\emptyset\}##, then ##(X,\tau)## is a topological space called the Moore plane. Describe the closured and interiors of the subsets of ##X##.

    I am having a hard time solving this exercise. I don't know where to start the exercise, should I separate in cases? I've tried to look at the cases ##A## open, ##A## closed but I couldn't do anything. I would appreciate any suggestions.
     
    Last edited: Aug 13, 2014
  2. jcsd
  3. Aug 14, 2014 #2
    (a) I believe the definition of ##\tau## should be ##\tau=\{A \in \mathcal P(X): \forall x\in A, A \in \mathcal F_x\} \cup \{\emptyset\}## or perhaps ##\tau=\{A \in \mathcal P(X): \forall p\in A, A \in \mathcal F_p\} \cup \{\emptyset\}##, but that's just a minor quibble.

    (b) Regardless I disagree with the claim that ##\tau## is a topology on ##X##. It's clear (?) that the families ##\mathcal F_p## are collections of discs or (mostly) half-discs depending on whether ##p## is in the upper half-plane or on the ##x##-axis. That means that most all of the ##A\in\tau## are discs and half-discs (and those that aren't are singletons on the ##x##-axis). It's not too difficult to see that his collection is not closed under intersections.
     
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