Describing closures and interiors of subsets of Moore Plane

1. Aug 13, 2014

mahler1

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. Aug 14, 2014

gopher_p

(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.