A problem on locally compact spaces in main chat room:

BTW some posts on the main:

Local compactness is preserved under continuous open onto mappings

Image of locally compact space under a continuous open map

@Secret So you are asking whether order topology on any ordinal $\alpha$ is totally disconnected?

yeah, and in fact, naively, I am wondering whether all well ordered sets are totally disconnected since the fact that we can always find the minimum element for every subset, should mean they are all singletons and thus any two elements can never be partitioned in terms of union of disjoint open sets?

My attempt so far is to choose $(0,\beta)\cup (\beta,\infty),\beta \in \alpha$, analogous to how total disconnectedness is proved for the rationals, but then I always miss 0 and $\beta$

So you are saying basically this: If we have any set $A$ and $m=\min A$, then both $\{m\}$ and $A\setminus\{m\}$ are clopen in $A$?

Hence if $A$ has at least two elements, then it is not connected.

@Secret Why don't you take $[0,\beta+1)$?

For order topology, the sets of the form $[0,x)=(-\infty,x)$ belong to the standard base, too.

ah, I thought too much about "what should I do with all the negative numbers"

I suspect that is what I try to express when you elaborated in term of the set $A$

right, if I use $[0,\beta+1)\cup (\beta,\infty)$ then I have successfully partition $\alpha$ into two disjoint open subsets thus (I think I need to double check to ensure I did not mix up totally disconnected with connected)

Not just that. You can partition any subset $A\subseteq\alpha$ in such way. (If $A$ has at least two elements.)

I see, thus since I can do the same partition throughout all ordinals and subsets of them, it follows that all well ordered sets are totally disconnected if it has 2 or more elements

Yes, I believe it's correct.

hmm, now it makes me wonder, whether the class of all ordinals is also totally disconnected since it is well ordered, but I need to check whether various notions of topology can generalise to proper classes

Here is a question on the main, but it did not get much response: Topology in a proper class.

Almost at the same time :-)

yeah, had to be very careful when playing with these really huge objects

Hello. I am working on the following problem: Show that $[0,1]^\omega$ is not locally compact in the uniform topology. I have been able to reduce the problem to showing that $\overline{B}(0,\epsilon)$, the closure of the $\epsilon$-ball, is not compact. The uniform topology is induced by the metric $\rho(x,y) := \sup \{ \min \{1, |x_i-y_i| \} ~|~ i \in \Bbb{N} \}$.

Since we are dealing with a metric space, I compactness of closed $\epsilon$-ball is equivalent to limit point compactness and sequential compactness. I having trouble deciding which route to take to obtain a contradiction. In the one case, I need to find an infinite set in $C = \overline{B}(0, \epsilon)$ that has no limit point, or I need to find a sequence in $C$ that has no convergent subsequence. However, both seem equally difficult. I could use a hint.