In at least one book and one Wikipedia article, I've seen someone specify which sequences are to be considered convergent, and what their limits are, and then claim that this specification defines a topology. I'm assuming that this is a standard way to define a topology. I want to make sure that I understand it.(adsbygoogle = window.adsbygoogle || []).push({});

Some of my thoughts: Suppose that we somehow specify all the convergent sequences in a set X, along with their limits, in a way that ensures that all subsequences of a convergent sequence have the same limit. Let F be the set of sequentially closed subsets of X. Then it's easy to show that the following holds: [itex]\emptyset,X\in F[/itex]. Every intersection of members of F is in F. Every finite union of members of F are in F. (We need the requirement about subsequences to prove that last one). So now we define [itex]\tau[/itex] as the set of subsets of X whose complements are in F, and use de Morgan's laws to show that it's a topology. Then we show that if the original specification says that [itex]x_n\rightarrow x[/itex], then every open neighborhood of x contains all but a finite number of terms of the sequence.

Looks good so far, but then I noticed that many different specifications give us the discrete topology, and that when we go in the other direction (i.e. determine all the sequences that are convergent with respect to a given topology), the cofinite topology gives us one of those specifications. Since the cofinite topology is a subset of the discrete topology, I'm thinking that my idea gives us thelargesttopology that ensures that all the sequences that were specified as convergent, are convergent with respect to the topology. Maybe we want the smallest?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Limits first, then topology.

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**