Some introductory Topology questions

[1MileCrash]

The problem is that you're working with finite topological spaces. Things are going to make more sense if you consider infinite topological spaces such as $\mathbb{R}^2$. Try to work with my suggestion in post 84. This is going to make more sense to you than finite topological spaces.

Oh, okay. I'll try doing that. It usually helps me a lot to look a little simple examples I construct, to see why the theorems are true and "witness them" in that way, so it's a bit challenging when all of these simple examples I can do seem rather "degenerate."

But, just so we're super clear, when you say infinite topological spaces, what exactly does that mean? (is the set the topology is on of infinite cardinality, or is the topology of infinite cardinality, or are members of the topology of infinite cardinality?)

 

[WannabeNewton]

It means that the set should be countably infinite or, even better, uncountable. Topologies on finite sets are not rich enough for you to get anything fruitful out of, as far as examples go; this is why you keep running into confusions.

 

[micromass]

Oh, okay. I'll try doing that. It usually helps me a lot to look a little simple examples I construct, to see why the theorems are true and "witness them" in that way, so it's a bit challenging when all of these simple examples I can do seem rather "degenerate."

But, just so we're super clear, when you say infinite topological spaces, what exactly does that mean? (is the set the topology is on of infinite cardinality, or is the topology of infinite cardinality, or are members of the topology of infinite cardinality?)

Usually it means that the space $X$ is infinite, not that the topology is infinite. However, in this case, if $\mathcal{T}$ is finite, then the examples are still degenerate.

I know you don't like it, but topology really is most important when things are infinite. Unlike group theory or linear algebra, where we can see important concepts in the finite (finite-dimensional) case. Topology is very different.

 

[1MileCrash]

It's not that I don't like it, I just need to adapt. :)

Thanks again guys

 

[lugita15]

For example, you have no idea even what a generic open set in $\mathbb{R}^2$ looks like!

At least intuitively, an arbitrary open set in $\mathbb{R}^2$ seems like it would just be a countable union of regions bounded by piecewise contnuous functions. Is it more complicated than that?

 

[micromass]

At least intuitively, an arbitrary open set in $\mathbb{R}^2$ seems like it would just be a countable union of regions bounded by piecewise contnuous functions. Is it more complicated than that?

OK, but continuous curves in the plane can be very very wild. The difficulty in proving the Jordan curve theorem shows this.

But what I meant is open balls ad rectangles are very well-behaved objects. We know exactly how they look like. We certainly do have an intuition how open sets look like, but I don't think we can ever give an explicit description.

 

[lugita15]

OK, but continuous curves in the plane can be very very wild. The difficulty in proving the Jordan curve theorem shows this.

Point taken; for instance space-filing curves. But is my characterization of open sets correct?

 

[micromass]

Point taken; for instance space-filing curves. But is my characterization of open sets correct?

I don't see any obvious counterexamples. But I can't really produce a proof at this moment.

 

[lugita15]

I don't see any obvious counterexamples. But I can't really produce a proof at this moment.

I just started a thread about it here, where I try to phrase the question more formally.