Advance Calculus Help! B(r,a) Help !?
I am a bit new to Topology, and I have a problem i am not sure how to sketch. I have a ball of B(r,a) = B(1,a)
which a = (3,5)
How would I go about drawing this? And justification please?
Any help is apprecaitioned! Im really confused!
- mcbengtLv 78 years agoFavorite Answer
Unfortunately, the notation B(r,a) has no standard meaning (even in the context of topology) but it is probably some version of the "ball with center a and radius r". Since a is (3,5), I'm imagining that this ball is supposed to be taking place in R^2 (the set of ordered pairs of real numbers--- since a = (3,5) is definitely an element of that set).
There are still two remaining ambiguities:
(1) To define balls, one usually fixes a "metric" (a way of measuring distance between points) and although there is sort of a "standard" choice of a metric in R^2, this choice is by no means the *only* choice.
But, for definiteness' sake, let's assume it's the usual metric. In which case, B(1, (3,5)) is the ball with center (3,5) and "radius" 1 in the usual sense of the word. (If you use a different metric on R^2, the "radius" of the ball is not a radius in the usual geometric sense. But with the standard metric it is.)
(2) The remaining ambiguity is whether the "ball" of radius 1 includes points that are a distance exactly 1 away from (3,5). This is just something you will have to look up in the definition of B(r,a): is it the set of all points whose distance to a is _less than or equal to_ r, or the set of all points whose distance to a is _less than_ r.
In the first case, you might indicate the ball by drawing the circle with center (3,5) and radius 1 with a dotted line and shade its interior. In the second case you would draw the circle with center (3,5) and radius 1 with a solid line and shade its interior. I hope this helped.