# Advance Calculus Help! B(r,a) Help !?

Hi,

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)

Any help is apprecaitioned! Im really confused!

Thanks!

Relevance

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.