@BrianMScott Yes. But, I think that's bad (And, Mariano once pointed out). Since, connectedness is the property of a individual set, we must have a nicer definition.
On the other hand, connectedness intuitively says that a set can’t be broken into pieces in any natural way, so it’s not surprising that the definition involves subsets.
For many purposes I prefer a similar but slightly different characterization: $X$ is connected iff it is not the union of two disjoint, non-empty clopen subsets.
Another characterization: $X$ is connected iff every continuous function from $X$ to the two-point discrete space is constant.
In the interest of giving advanced notice: I'll need to be away for a few weeks, during which I have no way of accessing the Internet. I hope I can return at a better time for me. So, for now: sayonara, auf Wiedersehen, and whatnot.
@KannappanSampath $X$ is the union of two disjoint non-empty open sets iff $X$ is the union of two disjoint non-empty clopen sets, because such open sets are necessarily clopen.
@BrianMScott A pedagogical/notational doubt: Is it wise to define the notion of connectedness for every subset or for the space and then every subset when considered as a subspace?
Hi all. I'm studying for a manifolds exam, and I have a question - an open subset of a manifold is always a submanifold, but is it always regular (i.e. embedded)?
I feel like it should be, since it should just inherit both its topology and differentiable structure, so it'll have the subspace topology & the restricted smooth structure.
Well. I have to prepare a lecture on Whitehead's problem and I have to finish the paper in algebraic topology, which I first have to get myself to start.
I thought that the symbol was the Fourier transform of the differential operator. But a simple example like $P := a(x) \frac{\partial}{\partial x}$ applied to $f$ and then transformed yields $\widehat{Pf(x)}= \xi \widehat{a(x) f(x)}$ whereas the symbol is supposed to be $\xi a(x)$. What am I missing?
I know that, but sometimes I have a lot in the backlog that I can scroll back to, and other times I can only scroll back a little bit. I was wondering if you still had the offending TeX in your backlog, whereas I didn't.
$$\begin{array}{c} x \in cl_Y(E) \iff & \exists ~ \{x_n\} \subseteq E \subseteq Y \subseteq X ~\text{such that}~x_n \to x \in Y \subseteq X& \\&\Updownarrow& \end{array}$$
Try this $\small{\begin{array}{c} x \in cl_Y(E) &\iff & \exists ~ \{x_n\} \subseteq E \subseteq Y \subseteq X ~\text{such that}~x_n \to x \in Y \subseteq X \\&&\Updownarrow \end{array}}$
\begin{table} \begin{tabular}{c c c} $ x \in cl_Y(E) $ & $\iff$ & $\exists ~ \{x_n\} \subseteq E \subseteq Y \subseteq X ~\text{such that}~x_n \to x \in Y \subseteq X$ \\ & & $\uppdownarrow$ \end{tabular} \end{table}
@KannappanSampath Then you just need to add another & below, some \hspace{1cm} or something
Yeah, I have never used any of the menus in TexMaker, I just try to remember everything. Thinking about switching to some simpler less cluttered typewriting program.
@N3buchadnezzar Oh, I am in no mood to try it out right now. And, on this note, the table is at the top of the page preceding everything else before it. Strange!
@robjohn I know that since $ lim_{n \to \infty} \frac{\log n}{n} = 0 $ it implies $ \frac{1}{a} \lim_{n \to \infty} \frac{\log\left( n^a \right)}{n^a}$. But how does the latter imply that $\lim_{n \to \infty} \frac{\log(n)^a}{x} = 0$ ?
He asked Diophantine stuff on MO as well. Eventually Scott Morrison asked him to ask on meta.MO first, and eventually he told him to stop asking at all.
It's not quite clear to me... Since I don't know whether the identification of $\ell^{\infty}*$ with the bounded additive set functions on $\mathcal{P}(\mathbb{N})$ works without any choice.
Yes, but I can't say anything about the dual. With choice it's the power set of the continuum, without it but dependent choice it can be continuum, without any choice, I have no idea. At least continuum.
@AsafKaragila By the way, what question specifically do you want me to take care of?
@Matt After I started typing out a small note on dense sets, it has ever since grown into a repository of some results on connected sets. Would you mind looking at it?
