# Tagged Questions

73 views

### increasing sequence in $\omega_1$

I want to prove that a (countable) increasing sequence in $\omega_1$ converges to some point. Here is my idea: I first try to find the limit point. We know any point in the increasing sequence (say ...
48 views

### Prove $\omega_1$ is first countable

Given a well order (W,$\le$), where W is uncountable, and $\omega_1$:= {x$\in$W: only countably many y$\in$X s.t. y $\le$ x}, prove $\omega_1$ is first countable. I saw a proof saying that {(a,x]: a ...
57 views

### Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
44 views

### Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)}$ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
53 views

### Definition of $\omega_1$, comparing it to $2^\mathbb{N}$?

I'm taking an Intro to Topology class, and we just started defining ordinals. We defined finite ordinals as: \begin{align*} 0 & = \varnothing \\ 1 & = \{0\} \\ 2 & = \{0,1\} \\ & ...
98 views

### What is the cardinality of the open ordinal space $[0,\Omega)$ if we remove it's limit ordinals?

Let $\Omega$ be the first uncountable ordinal. and for any limit ordinal $\lambda < \Omega$ Let $U_\lambda$ an open neighborhood of $\lambda$ with the order topology. what is the cardinality of the ...
100 views

### Is any subset of the open ordinal space $[0,\Omega)$ $G_\delta$?

Consider the open ordinal space $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal. Can I say that every subset of $[0,\Omega)$ is $G_\delta$? If yes, does this imply that $[0,\Omega)$ is ...
73 views

### Uncountable compact sets in ordinal topology

The question I am interested in is the following: Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied with order topology. Prove ...
139 views

### Long line is connected and compact

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.
80 views

### $X = [ 0,\omega_1 )$ is maximal countably compact

(a) : why $X = [ 0,\omega_1 )$ is a first countable space? ($\omega_1$ is the first uncountable ordinal number.) (b) : is $X = [ 0,\omega_1 )$ maximal countably compact?
301 views

### show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant

show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant.Where $S_{\mathbb{\Omega}}$ denote the first uncountable ordinal. There is a hint that for each ...
163 views

### Continuous surjective functions $\omega_1 \to \omega_1$.

I am looking for nontrivial examples of surjective continuous functions from $\omega_1$ onto $\omega_1$ (with both $\omega_1$'s in the order topology). What sorts of properties must these functions ...
74 views

### Compactifications of limit ordinals

I thought I knew that but it seems I don't. Let $\alpha$ be a countable, limit ordinal $\alpha>\omega$. Give $\alpha$ its order topology. What is the Stone-Čech compactification of $\alpha$? Is ...
36 views

### Is the space $X$ in the class dual to the spaces with the Souslin property?

Recall that $X$ is in the class dual to the spaces with the Souslin property: For any neighbourhood assignment $\{O_x: x\in X\}$, there is a subspace $Y \subseteq X$ such that $c(Y)=\omega$ and ...
89 views

### How to prove $X$ is normal?

Let $X=[0,\omega_1]\setminus \{\omega_1\}=[0,\omega_1)$. I known every linear order space is hereditarily normal. However, I would like to know the easier proof by which we can conclude that $X$ is ...
75 views

### How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
106 views

### topological spaces $\omega_1$ and $\omega_1 + 1$?

What are these as topological spaces? A proof I'm reading is referring to these as counter examples and I can't figure out what they mean?
45 views

### The proof from Van Mill, et al to prove $\omega_1$ is dually discrete

Here is the proof from J. van Mill, V.V. Tkachuk, R.G. Wilson; Classes defined by stars and neighbourhood assignments, Top. Appl. 154 (2007), pp.2127–2134, MR2324924, link to prove $\omega_1$ ...
41 views

### What is the meaning finite minimal cover?

In this lemma, it mentioned finite minimal cover. My question is this: What is the meaning finite minimal cover? Thanks.
35 views

### Lindelöfness in $\omega_1$

Is every Lindelöf subspace of the ordinal space $\omega_1$ countable?
50 views

### Do we have $\overline{\bigcup\{A_\beta: \beta<\alpha\}}=\bigcup\{ \overline{A_\beta}: \beta<\alpha\}$

Let $\alpha$ be any ordinal, and for the family $\{A_\beta: \beta< \alpha\}$, we have $A_{\beta_1}\subset A_{\beta_2}$ when $\beta_1<\beta_2<\alpha$, then do we have ...
47 views

### A question on $\omega_2$

Let $X=\omega_2$ and $\tau=\{U\subset X: |X\setminus U|<\omega_2\}$ is the topology on $X$. Then could the space $X$ be star countable? If it is not, what about the space $Z=\omega_1$ and ...
72 views

### A question on $\omega_1$

For the liner order space $\omega_1$, does it have countable chain condition? (I kown it is not separable, because for any countable subset $A$ of $\omega_1$, there exists an ordinal $\kappa$ ...
40 views

### How to show $\alpha^n$ is star countable for any ordinal $\alpha$?

How to show $\alpha^n$ is star countable for any ordinal $\alpha$? A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open cover of $X$, there is a countable ...
73 views

### How to understand closed subsets of limit ordinal?

On page 20, Constructibility, K.J.Devlin, (Let $\alpha$ is a limit ordinal)A set $A \subseteq \alpha$ is closed, iff $\bigcup A \cap \gamma \in A$ for all $\gamma < \alpha$. Equivalently, if we ...
95 views

### open subsets of $\omega_1+1$

Let $ω_1$ is the first uncountable ordinal and $ω_1+1$ together with the order topology. Let $U$ and $V$ two disjoint uncountable set consisting of non-limit ordinals, then $U$ and $V$ are open and ...
52 views

### $\omega_1$ is $C^*$-embedded in $\omega_1+1$

consider the space $ω_1$ (the first uncountable ordinal) and $ω_1+1$ together with the order topology. is it true that $ω_1$ is $G_\delta$-dense subspace of $ω_1+1$? $\omega_1$ is $C^*$-embedded in ...
77 views

### Existence of a $\sup$ and local compactness in the countable ordinals $\Omega$

I understand how the set of countable ordinals $\Omega$ is not compact, but how is it locally compact? Also, how can the existence of a least upper bound ($\sup$) for a non-decreasing sequence ...
89 views

### Is $\omega_{\alpha}$ sequentially compact?

For an ordinal $\alpha \geq 2$, let $\omega_{\alpha}$ be as defined here. It is easy to show that $\omega_{\alpha}$ is limit point compact, but is it sequentially compact?
85 views

### Equivalent definition of a closed set.

I try to prove the equivalence between "C is closed in an ordinal $\alpha$" and "each strictly increasing sequence of elements of C of length $< cf\alpha$ converge in C". For $\Rightarrow$, it's ...
182 views

### Limit ordinals vs. Points at infinity

The point at infinity $\infty$ makes $\mathbb{R}$ into a topological circle, i.e. into the real projective space $\mathbb{R}P$: $\infty$ enhances the original structure ($\mathbb{R}$) and makes it ...
685 views

### What does order topology over Ordinal numbers look like, and how does it work?

Space of ordinal numbers are one of the favorite examples of my professor in general topology. I quite understand the idea at the base of ordinal numbers (few things or nothing about the concept of ...
108 views

### Real line, field of real numbers and $\omega_1$ topological space difference

Real line is separable. Then, why is $\omega_1$ topological space not separable? IF this is true, doesn't this settle continuum hypothesis? Also, does the field of real numbers have anything to do ...
74 views

### Mappings from $\omega_1$

We've been studying various properties of $\omega_1$ (equipped with the order topology), and I recently came across these questions. Can anyone help? If $f$ maps $\omega_1$ onto a metric space $X$ ...
618 views

### Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
155 views

### countable product of $\omega_1 + 1$

How can I prove that the topological countable product of $\omega_1 + 1$ (with the order topology) is the union of $\omega_1$ closed nowhere dense sets?
367 views

### Closed Intervals of $\omega_1$ Are Compact

Consider the space $\omega_1$ (the first uncountable ordinal) together with the order topology (a convenient base for which is $\{(\alpha, \beta] \mid \alpha, \beta \in \omega_1\} \cup \{0\}$ ). I ...
115 views

### Continuous functions on $\omega_1$

Let $\lambda<\omega_1$ be a limit ordinal and consider the set $X=\{\alpha+1\colon \alpha<\lambda\mbox{ is a limit ordinal}\}$ Is the characteristic function $\chi_X$ continuous on $\omega_1$ ...
342 views

### How do you find the smallest of homeomorphic ordinals?

I am trying to get a better feel for the topology of ordinals and just received a great answer to this question where the Cantor-Bendixson rank and degree turn out to be a complete homeomorphism ...
I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can ...
I am not that familiar with arguments using ordinals so this may be pretty simple. Let $T$ be a topology on a set $X$. For each ordinal $\alpha$ let $T_{\alpha}$ be a topology on $X$ satisfying ...