4

In $X = \mathbb R^n$ (finite $n$ to make it simple), we can use either the open balls with the Euclidean metric, $B_d(x, \epsilon)$, or the Box topology (i.e., $n$-dimensional rectangular prisms with finite or infinite sides), as the basis for a topology $\mathcal T$, and these generate the same topology.

We can see this because for any open set $U \in \mathcal T$, for any $x \in X$, we can find a basis element $B$ (from either of these) such that $x \in B$ and $B \subset U$. This picture from Munkres illustrates it well: enter image description here

This makes enough sense to me. But I can't shake the feeling that they're still different, because they have different "shapes" (i.e., circles vs rectangles).

For example, another way to generate a topology from a basis is by taking the collection of all unions of the basis elements. This paints a different picture of generating the topology, where for any open set of the topology, we must be able to form it by taking a (possibly infinite) union of basis elements.

However, certain open sets of the topology will be formed trivially, or with the union of very few basis elements from one basis, while it may take the union of infinite basis elements from another basis. For example, to produce a rectangle with the "circles" basis, we would need infinite circles of decreasing diameter to get the corner or sides: enter image description here

But obviously only one from the "rectangles" basis. Is there any significance to this? or, is the significance that in topology, this really doesn't matter, and that's what's important?

skymonkey
  • 125
  • @JonathanZ Your comment is really an answer. Please post it as such. Maybe note too that the fact that the box has corners is also not a topological concept. – Ethan Bolker Nov 15 '23 at 22:33
  • @EthanBolker - Done. Don't know that I covered the idea of the corner well. Feel free to improve my answer. – JonathanZ Nov 15 '23 at 23:05

1 Answers1

3

The fact that those shapes look different to you is a property of the metric. Topologies are remarkably insensitive to stretching and compressing, unlike our human eyes, which love to use distance information. It takes a lot of experience and training to learn to ignore distance and only consider "topological shape".

Also, as mentioned in a comment, the corners aren't really a topological phenomenon - you could take ellipses as your bases elements, and you would still need an infinite number of circles to fill in up to the edges of the ellipse.

JonathanZ
  • 10,615
  • Thanks, this is really helpful! So this is the type of thing that's just not really relevant for topology I guess? – skymonkey Nov 15 '23 at 23:24
  • 2
    Yup, kind of like the color of your car isn't relevant to how fast you go. 20th century math worked a lot on paring down what one assumed, and seeing what you could still deduce with the smallest possible set of assumptions. Topology throws out as much as possible, and it turns out there are a lot of things that are stay true, even with the minimalest idea of "shape". – JonathanZ Nov 15 '23 at 23:30