All Questions
Tagged with ordinals general-topology
134
questions
2
votes
1
answer
105
views
Taking the limit beyond infinity, with the ordinals
Imagine a function $f:X\to X$ and $x\in X$ (keeping $f$ and $X$ vague on purpose) and let's define
$u_1 = f (x)$
$u_2=f^2(x)=f(f(x))$
$u_n=f^n(x)$
Let's also assume that $\forall n, u_{n-1} \neq u_n $ ...
- 169
5
votes
1
answer
157
views
Is the Stone Cech compactification $\beta\omega$ of $\omega$ radial?
$\beta\omega$ is sequentially discrete, that is, every convergent sequence is eventually constant. Thus, since the space is not discrete, the space is not sequential, that is, it's untrue that every ...
- 6,384
0
votes
0
answers
64
views
The range of any continuous function from $[0, \omega_1] \times [0, \omega_1] \rightarrow \mathbb{R}$ is countable
I came across this question:
“Prove that the range of any continuous function from $[0, \omega_1] \times [0, \omega_1] \rightarrow \mathbb{R}$ is countable”
I’m quite new to ordinals and I’m having a ...
- 309
0
votes
1
answer
146
views
Classification of compact Hausdorff spaces $X$ of cardinality $2^{\aleph_0}>|X|\geq \aleph_1$?
On my own time, I've been curious/investigating some properties of compact Hausdorff spaces of cardinality $\aleph_{\alpha}$, for $\alpha \geq 0$ a particular ordinal. I am aware of the following ...
- 5,830
0
votes
1
answer
70
views
Cantor-Bendixson rank of ordinal number
I need prove that:
There is a single limit point of type (o Cantor-Bendixson rank) $\alpha$ in $\omega^\alpha + 1$.
I try this:
Since $\omega^\alpha + 1 = \omega^\alpha \cup \{ \omega^\alpha \}$, ...
- 133
1
vote
1
answer
51
views
Order topology in $\omega^\alpha + 1$ and dynamical systems
Let $f \colon \omega^\alpha + 1 \to \omega^\alpha + 1$, where $\alpha \geq 1$ a continous funciton such that, exists $w \in \omega^\alpha + 1$ with $\mathcal{O}_f(w)$ dense in $\omega^\alpha + 1$.
The ...
- 133
1
vote
0
answers
38
views
Easier proof of homeomorphism classes of ordinals
I found a very interesting paper by V. Kieftenbeld and B. Löwe, "A classification of ordinal topologies" which discusses how to classify ordinals according to homeomorphism when they are ...
- 4,500
4
votes
0
answers
103
views
Prove that any continuous function $f:S_\Omega \rightarrow \mathbb{R}$ is eventually constant.
Let $\Omega$ be the first non-numerable ordinal number ( $\aleph_1$ is the first cardinal number greater than $\aleph_0$ when treated as an ordinal number is denoted by $\Omega$ ) and let $[0,\Omega)$ ...
- 147
0
votes
0
answers
169
views
Cantor-Bendixson derivative sets
I try to show that a compact subset $A\subset \mathbb{C}$ is at most countable if and only if there exists a countable ordinal number $\alpha$ (i.e $\alpha <\omega_{1},$ where $\omega_{1}$ is ...
- 51
0
votes
0
answers
31
views
Dense Compact subset of $X = ((\beta(D) \times (\mathfrak{c}^+ + 1)) \setminus (\beta(D) \setminus D) \times \{\mathfrak{c}^+\})$
Let $D$ be a discrete space of cardinality $\mathfrak{c}$ and $\beta(D)$ is the Stone Cech compactification of $D$. Let
$$
X = ((\beta(D) \times (\mathfrak{c}^+ + 1)) \setminus (\beta(D) \setminus D) \...
- 441
3
votes
2
answers
153
views
$\omega_1$ is not Lindelof
I am looking to prove that $\omega_1$ is not Lindelof. Here is my proof so far:
I am attempting to reveal a contradiction.
Consider the collection $\mathscr{U} = \{(a,b]: a,b \in \omega_1$, $a<b$, $...
- 2,782
1
vote
0
answers
48
views
$ \mathbb{X}=\left(\left[0,\omega_{1}\right]\times\left[0,\omega\right]\right)\setminus\left\{ \left(\omega_{1},\omega\right)\right\} $ is not T4
I need to show that the subspace $\mathbb{X}=\left(\left[0,\omega_{1}\right]\times\left[0,\omega\right]\right)\setminus\left\{ \left(\omega_{1},\omega\right)\right\} $ of $\left[0,\omega_{1}\right]\...
- 11
0
votes
0
answers
124
views
Show that omega_1 is normal?
I've already proven the following statement. If $A$ and $B$ are disjoint closed subsets of $\omega_1$, then at least one of them must be bounded. This proof is supposed to be used to prove that $\...
- 11
0
votes
1
answer
443
views
Questions about $\omega_1$ as a space (The Set of All Countable Ordinals)
The Set of All Countable Ordinals, $\omega_1$
I'm trying to understand several things regarding $\omega_1$ and trying to get a better intuition. I have four questions regarding this space (in bold), ...
- 2,452
1
vote
1
answer
108
views
Prove: If $A$ and $B$ are closed subsets of $[0,\Omega]$ then at least $A$ or $B$ is bounded
As usual, I am self studying topology and my knowledge of ordinals is meagre. Have
done some research on it.
Theorem 5.1 Any countable subset of $[0,\Omega)$ is bounded above.
(This exercise requires ...
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