Let Y denote the following subset of ( )

[Jamin2112]

Let Y denote the following subset of ... (URGENT!)

Homework Statement

Homework Equations

None.

The Attempt at a Solution

Halp! This is the last problem on my homework which is due tomorrow.


I don't know how to start this because, according to my textbook (Munkres), when we check whether a set A is open in Y, we are asking whether it belongs to the topology of Y. But I know nothing about the topology of Y, the subspace topology, because I first need to have a topology defined on ℝ² in order to have a subspace topology.

From the book:

Definition. Let X be a topological space with topology T. It Y is a subset of X, the collection

T_Y = { Y intsct U | U in T}

is a topology on Y, called the subspace topology.

 

[phiiota]

i haven't studied topology, but I've studied a little bit of metric spaces. do you use concepts like open balls to show a set is open?

 

[Jamin2112]

i haven't studied topology, but I've studied a little bit of metric spaces. do you use concepts like open balls to show a set is open?

Yes.

 

[phiiota]

okay, so for the first one, can you construct an open ball around any point in A that is contained in A? If you look at Y on a cartesian plane, you have all the points (x,y) where x>0, and y>=0. if you look at A, you have all the points where x>0, and y>0. for every point a=(xo,y0) in A, is there some r>0 st S(r)(a) is contained in A? (Sorry if my notation is a bit off). It looks like it. if you take r to be the min of d(x,0) and d(y,0), then shouldn't that ball be contained in A?

alternatively, what about it's complement? if it's complement is closed, then your set is open, right?

btw, there are relevant equations. what is the definition of open and closed?

 

[Jamin2112]

okay, so for the first one, can you construct an open ball around any point in A that is contained in A? If you look at Y on a cartesian plane, you have all the points (x,y) where x>0, and y>=0. if you look at A, you have all the points where x>0, and y>0. for every point a=(xo,y0) in A, is there some r>0 st S(r)(a) is contained in A?

Yes.

And when I look at the complement of A to see whether A is closed, I'm looking at Y\A is see if for every x in Y\A there exists an open ball containing x and contained in A. Right?