## general topology – A \$sigma\$-compact but not hemicompact space?

All spaces are at least Hausdorff. A topological space $$X$$ is called

• $$sigma$$-compact if there is a countable sequence $$(K_n)_{n of compact subsets of $$X$$ such that $$X=bigcup_n K_n$$.
• hemicompact if there is a countable sequence $$(K_n)_{n of compact subsets of $$X$$ such that for every $$Ksubseteq X$$ compact there is $$ninomega$$ with $$Ksubseteq K_n$$.

In particular a hemicompact $$T_1$$-space $$X$$ is $$sigma$$-compact since for every $$xin X$$ there is $$n$$ with $${x}subseteq K_n$$, hence $$X=bigcup_n K_n$$. I’m interested in conditions on $$X$$ that are sufficient to reverse this implication, but I am more interested in an example of a space $$X$$ (with $$X$$ at least Hausdorff, better if completely regular) which is $$sigma$$-compact but not hemicompact. I have checked the standard sources (Counterexamples in Topology and the pi-base website) but there are no examples of such spaces there, hence my question:

What is an example of an Hausdorff space $$X$$ which is $$sigma$$-compact but not hemicompact?

## usa – Can I drive up to the General Sherman (sequoia tree) general parking lot in a 30-foot RV and can I park it there during COVID-19 pandemic?

On the park website, it says that parking facilities are still open:

(…)

Roads and parking lots. Please park only in designated parking spaces. If a parking lot is full, refer to the park newspaper for other options.

The parks veichles length restrictions page states that vehicles longer than 24 feet are not advised.

You can not park next to the tree anyway unless you have a disability parking placard:

Those with disability parking placards can park in a small lot along the edge of the Generals Highway. From there, a wheelchair-accessible trail leads a short distance to the tree. If you don’t have a placard but can’t manage the Main Trail, during shuttle season you can ride park shuttles (all are wheelchair accessible, and some kneel) to the accessible trail.

(source)

## Solution of general linear stochastic differential equation system

The form of the general linear SDE system is the following:

$$dunderline{X}(t) = left(underline{underline{A}}(t) cdot underline{X}(t) + underline{a}(t) right) cdot dt + sum_{i = 1}^{I} left(underline{underline{B_i}}(t) cdot underline{X}(t) + underline{b_i}(t) right) cdot dW_i(t)$$

where $$underline{underline{A}}$$ and $$underline{underline{B}}$$ are matrices, $$underline{X}$$, $$underline{a}$$ and $$underline{b}$$ are vectors.

I would like to know the general solution of the $$underline{X}(t)$$, $$mathbb{E}left( underline{X}(t) right)$$ and $$mathbb{Var}left( underline{X}(t) right)$$. It is clear that $$mathbb{E}left( underline{X}(t) right)$$ is the solution without any $$dW_i(t)$$ noise. It is clear too that $$mathbb{Var}left( underline{X}(t) right) = mathbb{E}left((underline{X}(t)-mathbb{E}left( underline{X}(t) right)) cdot (underline{X}(t)-mathbb{E}left( underline{X}(t) right))^T right)$$.

But what are the exact formulas of the fields above? Do they even exist? If there is no exact formula in general case, is there any in (1) $$underline{underline{B_i}}(t) = 0$$, (2) $$underline{b_i}(t) = 0$$ or (3) $$I = 1$$ cases? What are they then?

Thank you very much for the answers!

## general topology – Using Arzela-Ascoli to prove there’s a converging sub-sequence

I am trying to solve the following problem in preparation for an upcoming exam in introductory topology and I’m unable to complete it entirely. My problem is with (b), but I’ve stated the full problem since it is likely that (a) is required for (b):

Let $$X$$ be a compact metric space.

a. Prove that every continuous function $$f:Xrightarrow mathbb{R}$$ is uniformly continuous.

b. Let $$T,S : X rightarrow X$$ be isometries.
Let $$f_1 in C(X,mathbb{R})$$. Denote $$f_{n+1} = frac{1}{2}Big(f_n(T(x))+f_n(S(x))Big) quadforall xin X, nin mathbb{N}$$ .
Prove that $${f_n}_{nin mathbb{N}}$$ has a converging subsequence using the sup norm.

So I’ve tried the following two approaches:

1. Use Arzela-Ascoli by first proving $$F:= {f_n}$$ is closed, bounded and equicontinuous. It seems impossible to prove it is closed – maybe I’m missing something? (tried assuming by contradiction there is a $$g$$ in $$cl(F)$$ not in $$F$$ but this didn’t take me too far).

2. Proving the space is complete (since $$X$$ is compact $$C(X,mathbb{R})= B_c(X,mathbb{R})$$ and using the fact that $$X$$ is a topological space and $$mathbb{R}$$ is complete) and then trying to prove that $${f_n}_{nin mathbb{N}}$$ is Cauchy, therefore converges, therefore has a convergent subsequence. Also got stuck in this case, although it seems more promising due to the isometries, but not sure.

What am I missing? Any help would be much appreciated!

## How should I handle Azure SQL hanging when scaling up from General Purpose to Hyperscale?

I’m in the process of scaling up an Azure SQL database from General Purpose to Hyperscale. This has been running for more than 12 hours. When I check the “Ongoing operations” it says that it is “Scaling database performance” and “Progress: 0%”.

I’m not sure if I should wait for it to complete, or click the “Cancel this operation” and try another approach. How should I handle Azure SQL hanging when scaling up from General Purpose to Hyperscale?

## general topology – \$X^{star} times Y^{star}\$ is \$KC\$ space, then \$Xtimes Y\$ is a \$k\$-space. My attempt…

The following theorem is found in the article “ON KC AND k-SPACES, A. García Maynez. 15 No. 1 (1975) 33-50”

Theorem 3.5: Let $$X, Y$$ be topological spaces. If $$X^{star} times Y^{star}$$ is KC then $$Xtimes Y$$ is a $$k$$-space

Definitions:

1. $$X$$ is a $$KC$$ space if every $$Ksubset X$$ compact is closed.
2. Let $$A⊂X$$. Then $$A$$ is a $$k$$-closed if for all $$K⊂X$$ compact it happens that $$A∩K$$ is closed in $$K$$
3. $$X$$ is a $$k$$-space if every $$k$$-closed set of $$X$$ is a closed set in $$X$$.
4. Let $$Csubset X$$. Then $$C$$ is compactly closed if for all $$Ksubset X$$ compact and closed, $$Ccap K$$ is compact.

The article refers that the theorem 3.5 is a consequence of theorem 3.4

Theorem 3.4: Let $$X,Y$$ be non-compact spaces and let $$Y^{star}=Ycup { infty }$$ be the one-point compactification of $$Y$$. Assume $$Y$$ is a $$KC$$ space. Then a set $$Csubset Xtimes Y$$ is compactly closed in $$Xtimes Y$$ if and only if $$Ccup (Xtimes { infty })$$ is compactly closed in $$Xtimes Y^{star}$$.

My attempt:
Let $$Asubset Xtimes Y$$ $$k$$-closed. I want to show that $$A$$ is closed in $$Xtimes Y$$.

By hypothesis and $$KC$$ is a hereditary property, we have to $$Xtimes Y$$ is $$KC$$ space. Also $$KC$$ is a factorizable property, then $$Y$$ is a $$KC$$ space. Now, clearly every $$k$$-closed set is compactly closed, therefore $$A$$ is compactly closed in $$Xtimes Y$$. Then by the theorem 3.4 we have $$Acup (Xtimes {infty })$$ is compactly closed in $$Xtimes Y^{star}$$.

Let $$Ksubset Xtimes Y$$ compact and closed. Consider the projection function $$rho_2 : Xtimes Y rightarrow Y$$, which is continuous and by the compactness of $$K$$, then $$rho_2 (K)$$ is compact and closed in Y, beacuse $$Y$$ is $$KC$$. Note that $$Y-rho_2 (K)$$ is open in $$Y^{star}$$ and $$Y^{star}-rho_2 (K)=Y^{star}cap (Y-rho_2 (K))$$, which is open in $$Y^{star}$$, therefore $$rho_2 (K)$$ is closed in $$Y^{star}$$ and $$Xtimes rho_2 (K)$$ is closed in $$Xtimes Y^{star}$$. As $$Ksubset Xtimes rho_2 (K)$$ we have $$K$$ is closed in $$Xtimes Y^{star}$$

Now remember $$Acup (Xtimes {infty })$$ is compactly closed in $$Xtimes Y^{star}$$ it happens that $$Acap K=Kcap (Acup (Xtimes {infty }))$$ is compact in $$Xtimes Y$$. Therefore $$Acap K$$ is closed in $$Xtimes Y$$ because $$Xtimes Y$$ is KC.

I still don’t have a clear idea of how to conclude that $$A$$ is closed in $$Xtimes Y$$. I hope you can help me or make any observations of my proof

Thank you.

## Prove of the recurrence from general recurrence

We have general recurrence for A284005
$$a(n)=(1+b(n))a(leftlfloorfrac{n}{2}rightrfloor), a(0)=1$$
where $$b(n)$$ (A000120) is number of $$1$$‘s in binary expansion of $$n$$
$$b(n)=b(leftlfloorfrac{n}{2}rightrfloor)+nbmod 2, a(0)=0$$
there exist another recurrence
$$a(n)=2a(f(n))+sumlimits_{k=0}^{leftlfloorlog_2{n}rightrfloor-1}a(f(n)+2^k(1-T(n,k))), a(0)=1, a(1)=2$$
where $$f(n)$$ (A053645) is distance to largest power of $$2$$ less than or equal to $$n$$
$$f(n)=n-2^{leftlfloorlog_2{n}rightrfloor}$$
$$T(n,k)=leftlfloorfrac{n}{2^k}rightrfloor operatorname{mod} 2$$
Is there a way to prove recurrence with a sum using general recurrence?

## general topology – Examples of homeomorphic spaces

Are the following pairs of spaces homeomorphic?

(i) $$mathbb Q cup (0, 1)$$ and $$mathbb R setminus mathbb Q$$ (as subspaces of $$(mathbb R, tau_{usual}))$$;

(ii) $$(mathbb C, tau_{disc})$$ and $$(mathbb C, tau_{cocountable})$$;

(iii) a circle with one point removed (in $$mathbb R^2$$ with $$tau_{usual}$$) and $$(mathbb R, tau_{usual})$$;

(iv) $$mathbb N$$ with topology $${1, 2}, {3, 4}, {5, 6}, {7, 8}, ldots$$ and all unions of these, and $$mathbb N$$ with topology $${1, 3}, {2, 4}, {5, 7}, {6, 8}, {9, 11}, {10, 12}, ldots$$ and all unions of these.

The following have answers

(v) $$(mathbb R, tau_{cocountable}), (mathbb R, tau_{cofinite})$$

(vi) $$mathbb R^2$$ and the surface of a sphere with one point removed (natural metric
topologies here)

(vii) $$(mathbb Q,l_x), (mathbb Q, l_y)$$, where $$x, y$$ are two distinct rational numbers and $$l_p$$ is included-point topology

Book’s answers to (v) and (vii):

(v) No. The statement ‘there is a countably infinite closed subset’ is
(obviously) a homeomorphic invariant, is true in $$(mathbb R, tau_{cocountable})$$ and is false in $$(mathbb R, tau_{cofinite})$$

(vii) Yes. The map $$h : mathbb Q → mathbb Q$$ given by $$h(x) = y, h(y) = x, h(z) = z$$
when $$z ne x$$ or $$y$$ is routinely checkable to be a homeomorphism.

My answers/questions regarding question above:

(i) $$mathbb Q, (0, 1)$$ are not compact and so their union is not compact meaning there’s no continuous inverse function from $$mathbb Q cup (0, 1)$$

(ii) Any function from a discrete space to an indiscrete space is continuous. Since idenity $$i$$ is continuous and bijective, $$i$$ should work here to show homeomorphism

(iii) I think this is a special case of Stereographic projection

(iv) Any inverse image of an open set in $$mathbb N$$ is a union of $$2$$-sets. Since the given $$2$$-sets are open, their union is open as well. This works in both directions, so a continuous function with its continuous inverse likely exists. I am not sure how to find a concrete homeomorphism, but would $$f(x, y) = (x, y – 1)$$ if $$y$$ is odd and $$f(x, y) = (x + 1, y)$$ if $$x$$ is even work?

Are my answers to (i) through (iv) correct? If not (or if incomplete), how do I improve them?

(v) This question might be trivial, but I am still new to the very basics of topology. Consider $$f: (mathbb R, tau_{cofinite}) to (mathbb R, tau_{cocountable})$$. Every $$Y in (mathbb R, tau_{cocountable})$$ must have a pre-image $$X in (mathbb R, tau_{cofinite})$$. By definition, every $$X$$ is open, but for $$f$$ to be continuous $$X$$ must be cocountable. Correct?

(vi) It’s yet another special case of Stereographic projection. Correct?

(vii) Is $$h(y)$$ a typo? Did they mean $$h^{-1}(y)?$$

Thanks.