Does it make geometric sense to say that open rectangles and open balls generate the same open sets

Question

I have always been bothered by when people say:

The open ball (i.e. $L_2$ ball) and the open rectangle (i.e. $L_\infty$ ball) generates the same open sets (topology) on $\mathbb{R}^2$

The proof is something of the sort you can always put a square inside a ball and a ball inside of a square...

But geometrically does it make sense?

I find it hard to believe that given a random "blob" in $\mathbb{R}^2$, it is generated by through countable union of open balls or open rectangles. I mean rectangles have corners don't they...?

How is it geometrically intuitive that every open set is generated by open balls or open rectangles?

Answer 1

Infinite Unions is the key to understand this. Intuitively, the situation is described in the following figure:

Answer 2

but geometrically does it make sense?

The only geometric thing to consider is that you can inscribe or circumscribe a circle with a square, and that all makes complete sense.

From there it is completely clear how one would express an open set (defined by circles) as a union of open squares: capture each point inside the set with an open circle, then shrink each one to an open square. The union of all those squares is inside the open set, and every point of the open set is contained in one.

The same argument works in the other direction. Shape is not all that important For open sets. After all, you don't even have a metric a lot of the time.

Answer 3

Remember that topoloy can be though of as the study of "stretching without tearing" and then it is common to identify two spaces when one can be deformed into the other (ie homeomorphism). Anyhow, suppose you've got a circle made of super strecthy material, then you can stretch it, without tearing, into a rectangle; and vice versa. Now if you've got an open set which is just the Union of a bunch of open circles and you know how to deform the circles into rectangles, then that open set is also a union of a bunch of open rectangles. :-) hope that helps!

Answer 4

Another thing to consider is what if I took a rectangle and placed the corner on the boundary of the circle. I could do this an infinite amount of times and eventually work my way all around the circle. The reason I can do this is because the circles and rectangles are both open.