2
votes
3answers
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 ...
0
votes
1answer
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 ...
5
votes
0answers
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 ...
3
votes
1answer
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 ...
0
votes
2answers
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\} \\ & ...
2
votes
2answers
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 ...
1
vote
1answer
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 ...
2
votes
1answer
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 ...
1
vote
1answer
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.
-1
votes
1answer
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?
2
votes
3answers
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 ...
7
votes
1answer
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 ...
2
votes
2answers
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 ...
1
vote
0answers
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 ...
1
vote
3answers
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 ...
3
votes
1answer
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 ...
3
votes
2answers
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?
2
votes
1answer
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 $\omega_1$ ...
2
votes
1answer
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.
3
votes
1answer
35 views

Lindelöfness in $\omega_1$

Is every Lindelöf subspace of the ordinal space $\omega_1$ countable?
3
votes
1answer
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 ...
2
votes
1answer
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 ...
4
votes
1answer
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 $\kappa$ ...
0
votes
1answer
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 ...
1
vote
2answers
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 ...
2
votes
1answer
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 ...
4
votes
1answer
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 ...
1
vote
1answer
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 ...
1
vote
2answers
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?
1
vote
1answer
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 ...
2
votes
1answer
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 ...
1
vote
2answers
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 ...
0
votes
2answers
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 ...
6
votes
2answers
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 $X$ ...
19
votes
3answers
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 ...
3
votes
1answer
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?
2
votes
3answers
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 ...
2
votes
1answer
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 $\omega_1$ ...
13
votes
1answer
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 ...
12
votes
3answers
932 views

Countable compact spaces as ordinals

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 ...
5
votes
2answers
111 views

Transfinite sequences of topologies

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 ...