Analysis WebNotes

Open Sets Demonstration

The purpose of this demo is to give you a chance to get a hands-on feel for a number of concepts related to the definition of open sets. The way they work, is you select a set to work with, and then click on a point in the set to generate an open disk centered on that point which lies entirely inside the open set.

In this page you need to make a number of choices for the set up of the exercise. If you ever want to come back and change those options you can follow the link at the bottom of the next page.

The first thing you need to do is choose the open set you want to work with. Click on the button for the shape of set you want (the default is "half-plane"):

Half Plane: Oval: Blob1: Blob2:

Or you can choose from this list to work with one of the same shapes, but in a larger size:

Half Plane: Oval: Blob1: Blob2:

There are two modes you can run this demo in. In regular mode you see the open set, you click on a point in it, and then the computer draws an open ball around your point. In battleships mode, you never actually see the open set. You click blindly on the image, and the computer draws in your open disk. You get to see all your previous disks, so by doing this repeatedly, you see the open set you selected filling out. (This illustrates the principle that an open set can always be expressed as the union of (possibly infinitely many) open disks.)

Regular: Battleships:

There are three ways that you can select the size of the open balls you use.

You can opt to input the radius you want in an input box (first, click on the set to select the point you're interested in, and then you'll be prompted for the radius).

Alternatively, you can have the computer choose a radius for you, and that can be either some random radius (random, but small enough to fit in the set), or it can be that largest, or optimal, radius that will fit in the set.

Now choose the method you want to use to find the radius:

Input radius: Random: Optimal:

When you're happy with your choices, press:

At any time you can return to this page by following one of the links at the bottom of the subsequent pages.