I don't like the way my current question was treated. [closed]

Question

Update: the main post is reopened.

I don’t like the way my last question was treated. All terms I used are standard and I got downvoted to -4 with no relevant answer.

Answer 1

3 of the 5 close votes came from people with tag badges in topology-type fields, so I think the "Needs details or clarity" reason is not because the terms are not standard. In any case, you should always edit new information into the post body, not merely post a comment. This way, no one else will think the same thing as the first person did.

Despite (now) looking ok at the "micro level" (Mathjax) and not so bad at the level of each sentence, I am actually having trouble parsing the overall meaning of your entire post. I think this is mainly because you say this is a problem, but from the quotations of the text ("and choose a real number $r$ in this intersection"), it looks like you are following part of a proof that is given, or at least, the full exercise is not given. You also say "this I proved", but as currently written, there is nothing to "prove".

I think the easiest way to make your question much clearer would be to include a screenshot or transcribe the relevant part of Conover (not Connover, as you wrote in your comment.) From the phrase "and choose a real number $r$ in this intersection", I have found the relevant exercise in Conover, as Exercise 3 in Section 4.5 of Chapter 7

3 (This exercise requires a knowledge of ordinal numbers.) From Problems 1 and 2, it is immediate that the space of countable ordinals is pseudocompact. In fact, it is actually more than just pseudocompact because a continuous real-valued function on $[0, \Omega)$ is ultimately constant, so is more than just bounded. (A function $f$ on $[0, \Omega$ )is ultimately constant if there is an ordinal number $a \in[0, \Omega]$ such that $f(x)=f(y)$ for all $x, y \in$ $(a, \Omega)$.) Showing that every continuous real-valued function on $[0, \Omega)$ is ultimately constant is not easy.Use the following outline, which is adapted from Chapter 5 of Gillman and Jerison [6].

a) If $A$ and $B$ are closed subsets of $[0, \Omega)$, then at least oneof $A$ or $B$ is bounded (i.e., there is an ordinal number $a<\Omega$ such that $A \subset[0, a]$ or $B \subset[0, a]$,

b) Show that for each ordinal number $a \in[0, \Omega),[a, \Omega)$ is countably compact.

c) Show that the continuous image of a countably compact space is countably compact.

d) Let $f:[0, \Omega) \rightarrow \mathbf{R}$ be continuous and show that for each ordinal number $a \in[0, \Omega), f(a, \Omega)$ is a compact subset of $\mathbf{R}$ (see Theorem 4.1). Deduce that $\cap\{f(\mathrm{a}, \Omega): \mathrm{a} \in[0, \Omega)\} \subset \mathrm{R}$ is not empty, and choose a real number $r$ in this intersection. Show that for each $n \in \mathbf{Z}^{+}$, the set $\{x \in[0, \Omega): f(\mathrm{x}) \geq$ $1 / n=S_{n}$ is closed in $[0, \Omega)$ and is disjoint from $\boldsymbol{f}^{-1}(\{\boldsymbol{r}\})$ which is also closed in $[0, \Omega)$. Using (a) and the properties of $r$, showthat $S_{n}$ is bounded and let $\mathrm{fl}_{\mathrm{n}} \in[0, \Omega)$ be such that $S_{n} \in\left[0, a_{n}\right)$. Use Section 5 of Chapter 2 to get an ordinal number $a \in[0, \Omega)$ such that $a_{n} \leq a$ for all $n \in \mathbf{Z}^{+}$, and show that $f(x)=r$ for all $x \in(a, \Omega)$. Conclude that $f$ is ultimately constant, so every real-valued continuous function on $[0, \Omega)$ is ultimately constant. In particular, $[0, \Omega)$ is pseudocompact.

source that you can copy into your Question:

>3 (This exercise requires a knowledge of ordinal numbers.) From Problems 1 and 2, it is immediate that the space of countable ordinals is pseudocompact. In fact, it is actually more than just pseudocompact because a continuous real-valued function on $[0, \Omega)$ is ultimately constant, so is more than just bounded. (A function $f$ on $[0, \Omega$ )is ultimately constant if there is an ordinal number $a \in[0, \Omega]$ such that $f(x)=f(y)$ for all $x, y \in$ $(a, \Omega)$.) Showing that every continuous real-valued function on $[0, \Omega)$ is ultimately constant is not easy.Use the following outline, which is adapted from Chapter 5 of Gillman and Jerison [6].
>
>a) If $A$ and $B$ are closed subsets of $[0, \Omega)$, then at least oneof $A$ or $B$ is bounded (i.e., there is an ordinal number $a<\Omega$ such that $A \subset[0, a]$ or $B \subset[0, a]$,
>
>b) Show that for each ordinal number $a \in[0, \Omega),[a, \Omega)$ is countably compact.
>
>c) Show that the continuous image of a countably compact space is countably compact.
>
>d) Let $f:[0, \Omega) \rightarrow \mathbf{R}$ be continuous and show that for each ordinal number $a \in[0, \Omega), f(a, \Omega)$ is a compact subset of $\mathbf{R}$ (see Theorem 4.1). Deduce that $\cap\{f(\mathrm{a}, \Omega): \mathrm{a} \in[0, \Omega)\} \subset \mathrm{R}$ is not empty, and choose a real number $r$ in this intersection. Show that for each $n \in \mathbf{Z}^{+}$, the set $\{x \in[0, \Omega): f(\mathrm{x}) \geq$ $1 / n=S_{n}$ is closed in $[0, \Omega)$ and is disjoint from $\boldsymbol{f}^{-1}(\{\boldsymbol{r}\})$ which is also closed in $[0, \Omega)$. Using (a) and the properties of $r$, showthat $S_{n}$ is bounded and let $\mathrm{fl}_{\mathrm{n}} \in[0, \Omega)$ be such that $S_{n} \in\left[0, a_{n}\right)$. Use Section 5 of Chapter 2 to get an ordinal number $a \in[0, \Omega)$ such that $a_{n} \leq a$ for all $n \in \mathbf{Z}^{+}$, and show that $f(x)=r$ for all $x \in(a, \Omega)$. Conclude that $f$ is ultimately constant, so every real-valued continuous function on $[0, \Omega)$ is ultimately constant. In particular, $[0, \Omega)$ is pseudocompact.

This makes it clear that the question is far outside my comfort zone so I cannot directly help you further write a clearer question, nor can I solve the exercise myself. But it is clear to me that some improvement can be made. My suggestions are this:

1. turn the $o$ and $O$s in the question to $0$s (zero)
2. Add the information you provided in the comments into the main question
3. Give the book name, author, section, problem number. And correctly (not Connover). You might want to use this: Conover, Robert A., A first course in topology. An introduction to mathematical thinking, Baltimore, Md.: The Williams & Wilkins Company. XIX, 245 p. $ 11.95 (1975). ZBL0296.54001.

<cite authors="Conover, Robert A.">_Conover, Robert A._, A first course in topology. An introduction to mathematical thinking, Baltimore, Md.: The Williams \& Wilkins Company. XIX, 245 p. $ 11.95 (1975). [ZBL0296.54001](https://zbmath.org/?q=an:0296.54001).</cite>

4. Write out your actual problem clearly. I would definitely consider including more or even all of the above exercise. Perhaps: pretend you are writing your own text book on topology, and are writing an exercise that is about your specific issue. How would you write it?
5. Really an extension of the above: do your best to answer any questions people might have before they even ask. In particular: make sure that someone who hasn't read Conover can also understand your problem.
6. Do the above in one edit (or 2); don't continuously edit your post over a large span of time.
7. Respond as quickly as you can to comments, and politely.

Hope the above helps.

Answer 2

I wanted to answer your question in main but I ended up searching in lots of the "mumble jumble" in order to find out what exactly is the question you are asking. (The current version that with lots of community efforts is getting better.)

When asking such questions, one should never assume that the readers know your book. Thinking that "All terms I used are standard" may often cause the failure of your question when

• your question depends heavily on the context of the book, particularly choices of notation and definitions;
• your question is from a sub-question of a lengthy original one that the author designs for students to understand a section of their book.

The question should be self-contained so that one does not need to walk to the library to borrow a copy of your book in order to understand your question (let alone answer it).

Also, for such questions, you should keep the distraction such as "I did a lot of googling and searching on MSE to solve them." away from your main narrative of the question in order to make it clean and clear.

Besides Calvin's many excellent pieces of advice, below is a version of the post that I would write if I were to ask the question.

[Post begins]

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Denote $\Omega$ as the smallest uncountable ordinal number and consider the space of countable ordinals $[0,\Omega)$. Let $f:[0,\Omega)\to\mathbf{R}$ be a continuous function.

(1) Show that for each ordinal number $a\in[0,\Omega)$, the set $f([a,\Omega))$ is a compact subset of $\mathbf{R}$.

(2) Deduce that the set$$\cap\{f([a,\Omega)):a\in[0,\Omega)\}\subset\mathbf{R}$$ is not empty.

This is one intermediate question from an exercise in Robert Conover's A First Course in Topology: Exercise 3) in Section 4.5 of Chapter 7. The original one is lengthy and I include it at the end of the post. The problem above is directly from 3(d) of the excerpt.

Question. How can I get (2) (from (1))?

[Your question would already have clear "context" as the source is mentioned at the beginning and also the excerpt below. If you want to add any thoughts here, make sure that they are really relevant to difficulties you have to get (2) from (1), instead of an irrelevant mumble jumble that disturbs the post.]

---

Context. The original problem in Conover's book:

3. (This exercise requires a knowledge of ordinal numbers.) From Problems 1 and 2, it is immediate that the space of countable ordinals is pseudocompact. In fact, it is actually more than just pseudocompact because a continuous real-valued function on $[0, \Omega)$ is ultimately constant, so is more than just bounded. (A function $f$ on $[0, \Omega)$ is ultimately constant if there is an ordinal number $a \in[0, \Omega]$ such that $f(x)=f(y)$ for all $x$, $y \in(a, \Omega)$.) Showing that every continuous real-valued function on $[0, \Omega)$ is ultimately constant is not easy. Use the following outline, which is adapted from Chapter 5 of Gillman and Jerison [6].

   a) If $A$ and $B$ are closed subsets of $[0, \Omega)$, then at least oneof $A$ or $B$ is bounded (i.e., there is an ordinal number $a<\Omega$ such that $A \subset[0, a]$ or $B \subset[0, a]$,
   b) Show that for each ordinal number $a \in[0, \Omega)$, $[a, \Omega)$ is countably compact.
   c) Show that the continuous image of a countably compact space is countably compact.
   d) Let $f:[0, \Omega) \rightarrow \mathbf{R}$ be continuous and

   • show that for each ordinal number $a \in[0, \Omega), f(a$, $\Omega$ ) is a compact subset of $\mathbf{R}$ (see Theorem 4.1).

   • Deduce that $\cap\{f(\mathrm{a}, \Omega): \mathrm{a} \in[0, \Omega)\} \subset \mathbf{R}$ is not empty, and choose a real number $r$ in this intersection.

   • Show that for each $n \in \mathbf{Z}^{+}$, the set $\left\{x \in[0, \Omega): f(\mathrm{x}) \geq 1 / n=S_{n}\right.$ is closed in $[0, \Omega)$ and is disjoint from $f^{-1}(\{r\})$ which is also closed in $[0, \Omega)$.

   • Using (a) and the properties of $\mathrm{r}$, show that $S_{n}$ is bounded and let $\mathrm{fl}_{\mathrm{n}}$ $\in[0, \Omega)$ be such that $S_{n} \in\left[0, a_{n}\right)$.

   • Use Section 5 of Chapter 2 to get an ordinal number $a \in$ $[0, \Omega)$ such that $a_{n} \leq a$ for all $n \in \mathbf{Z}^{+}$, and show that $f(x)=r$ for all $x \in(a, \Omega)$. Conclude that $f$ is ultimately constant, so every real-valued continuous function on $[0, \Omega)$ is ultimately constant. In particular, $[0, \Omega)$ is pseudocompact.

[Post ends]