# Set intersection Proof

• Feb 22nd 2010, 07:43 PM
jzellt
Set intersection Proof
Show that the intersection of two non-empty open sets is open.
Proof:
Let X and Y be two nonempty open sets. Let p e XnY (n = intersection) We must show that p is an interior point of XnY. Since p e XnY, by def. of intersection, p e X and p e Y.
• Feb 22nd 2010, 07:44 PM
Drexel28
Quote:

Originally Posted by jzellt
Show that the intersection of two non-empty open sets is open.
Proof:
Let X and Y be two nonempty open sets. Let p e XnY (n = intersection) We must show that p is an interior point of XnY. Since p e XnY, by def. of intersection, p e X and p e Y.

In what kind of space are we working? In some this is the DEFINITION of open sets, in others it is almost as easy.
• Feb 22nd 2010, 07:47 PM
jzellt
We're working in the complex plane
• Feb 22nd 2010, 07:56 PM
Drexel28
Quote:

Originally Posted by jzellt
We're working in the complex plane

Well, I don't know what you know or what kind of rigor this requires.

Either the two open sets are disjoint in which case their intersection is empty, one is a subset of the other in which case their intersection is themselves, or neither is true. In this case, this will be an ovaloid with no boundary, which is clearly geometrically "open".

That's all you get till I see some work!
• Feb 23rd 2010, 04:24 AM
HallsofIvy
So we are working, at least, in a metric space. (In a general topological space, the fact that the intersection of two open sets is open is part of the definition of "topology".)

If the intersection is empty it is, by definition, open.

If the intersection is not empty, let p be any point in the intersection. Then it is in both X and Y.

Since p is in X and X is open, there exist \$\displaystyle r_1> 0\$ such that the open ball (neighborhood) centered on p with radius \$\displaystyle r_2\$ is a subset of X.

Since p is in Y and Y is open, there exist \$\displaystyle r_2> 0\$ such that the open ball (neighborhood) centered on p with radius \$\displaystyle r_2\$ is a subset of Y

. Let r be the smaller of \$\displaystyle r_1\$ and \$\displaystyle r_2\$ and show that the neighborhood centered on p with radius r is a subset of \$\displaystyle X\cup Y\$.