closed open both or neither sets in metric space

Dec 2009
84
7
let $d((a,b),(c,d))=d_{0}(a,c)+|b-d|$ be a metric on $\mathbb R^2$


let
X=$\{(x,0):x\in \mathbb R \}$ with the induced metric $d_{X}$


Y=$\{(0,y):y\in \mathbb R \}$ with the induced metric $d_{Y}$


A=$\{(x,y):x^2+y^2\leq4\}$


B=$\{(x,y):x^2+y^2<9\}$




i have to decide if the following subsets of X and Y are open,closed, neither or both.


1) $A \cap X$ with respect to $d_{X}$


2) $B \cap X$ with respect to $d_{X}$




3) $A \cap Y$ with respect to $d_{Y}$


4) $A \cap Y$ with respect to $d_{Y}$


can someone check i have this correct.




ive said that $d_{X}$, the restriction of d to X is just the discrete metric $d_{0}$ so i have both 1 and 2 are open and closed.


and $d_{Y}$ is just the euclidean metric so 3 is closed and 4 is open
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey jiboom.

Have you tried setting a couple of variables to zero [which would happen in the dx and dy] and then testing that against the axioms? [Think of what happens when you take two constraints and combine them and set them consistent when the combination takes place].