Tuesday, March 01, 2005

The indiscrete topology

In Topology class on Monday, we were discussing whether certain sets were compact in certain topologies. Now, I'm okay with the product topology, the uniform topology, and the box topology. I'm okay with the discrete topology and the finite complement topology. But then someone mentions the indiscrete topology. So I'm wondering, just what is this indiscrete topology? I could not remember what it was. Reasoning: Maybe it's the opposite of the discrete topology. But no, the finite complement topology is kind of the opposite of the discrete topology. But I didn't want to ask, because everyone else seemed to know. After class, I looked it up. Do you know what the indiscrete topology is? No, you don't. It's just the empty set and the whole space. Is that absurd or what? I know -- topologies aren't allowed to have two names! The other name, and the more logical name, for that topology, is "the trivial topology." Because honestly, that is the most trivial topology you could ever think of. Honestly. You can write it in two symbols! It is the only topology that can make that not-so-lofty claim. Trivial.

So I'm going to come up with other names for the other topologies too. For instance, if I don't know the answer to a problem such as "which topology has such-and-such property?" on a test, I'll say, "the inconsiderate topology." Then if anyone says that is wrong, that the real answer was the box topology, I will simply say, "oh, 'the inconsiderate topology' is just my other name for the box topology, like the indiscrete topology, except that this one goes so far as to be inconsiderate." Because if the trivial topology can get a special name, I think every topology deserves a special name.