let d be the euclidean metric of $\mathbb R^2$
let
W={$(x,y):x^2+y^2<4$} $\mathbb U$ {$(x,y):x^2+y^2$ $<4 $ and $x^2+y^2\leq9$}
X={$(x,x):x\in \mathbb R$} $\mathbb U$ {$(x,-x):x\in \mathbb R$}
Y={$(q,1):q\in \mathbb Q$}
Z={$(x,y):(x+2)^2+y^2=4$} $\mathbb U$ {$(x,y):(x-2)^2+y^2<4$}
I have to decide if the above sets are d-connected or d-disconnected.
i'm inclined to say W is not connected as its the circle of radius 9 take away the circumference of circle radius 4. so for example i cant get to the origin from a point near edge of big circle.
But to prove this i need to write W as union of 2 disjoint sets that are d-closed and d-open.(
to do this according to my book i need to say, as W is subset of the reals any closed subset of W is intersect of W and F where F is a closed set in $\mathbb R^2$
hopefully ive understood the theory correctly,but now i'm confused how to find my subsets. (im tempted to say the set $S_{1}$={$(x,y):x^2+y^2<4$} is open as it is just an open subset of the reals and the set $S_2$= {$(x,y):x^2+y^2$ $\leq9 $}-{ $x^2+y^2<4$} is open in W because although the edge of the circle of radius of 9 is included in the set, i can put "balls" around these points as there are no points in W inside my "balls that go outside the edge of the circle. So W=$S_1$$\mathbb U$$S_2$ and this is a disjoint union. is this anywhere near correct?
i have the same trouble for the other sets. i'm guessing
X is connected as its just a cross
Y is not connected as i have gaps in the line through y=1 at each irrational
Z is connected as it is 2 circles that touch at the origin and the origin is included in the space Z
which leads me how do i show it is connected? do i have to show there are no open subsets of X whose union is the whole of X?
Does this work for W?
i have W={$(x,y):x^2+y^2<4$} $\mathbb U$ {$(x,y):x^2+y^2$ $<4 $ and $x^2+y^2\leq9$}
so could i let
S={$(x,y):x^2+y^2<4$}
T={$(x,y):4<x^2+y^2\leq 9$}
so W=S$\cup$T and S and T are disjoint
now
S = W $\mathbb n$ {$(x,y):x^2+y^2<4$}
and {$(x,y):x^2+y^2<4$} is open in $\mathbb R^2$
and can i say
T=W$\mathbb n${$(x,y):4<x^2+y^2<10$}
and as $\mathbb n${$(x,y):4<x^2+y^2<10$} is open in $\mathbb R^2$ then
T is open in W
the conclusion is then since i can write W as the union of two disjoint W open sets it is therefore disconnected?
im still unsure about the other sets, if someone could even just confirm i have the correct connected/disconnected for X ,Y, Z so i know im in the right area to try and proceed.
thanks