All Questions
Tagged with cardinals general-topology
132
questions
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2
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58
views
Cardinality of a set of disjoint open sub intervals of $( 0 ,1)$
Let $A$ be any collection of disjoint open subintervals of $(0 ,1)$ . Then what is maximum cardinality of $A$ ?
I know one easy way to prove its countable is that every open interval has rational ...
2
votes
1
answer
44
views
Singular cardinals and $\kappa$-Lindelöf spaces
Say a space is $\kappa$-Lindelöf provided that for every open cover of the space, there exists a subcover of cardinality $<\kappa$. So $\aleph_0$-Lindelöf is compact, and $\aleph_1$-Lindelöf is ...
- 6,384
2
votes
1
answer
59
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Does a separable, $US$, sequential space have cardinality at most the continuum?
Let $X$ be a separable sequential space with unique sequential limits ($US$). Can we prove that $X$ has cardinality at most $\mathfrak c=2^{\aleph_0}$?
Context.
If $X$ were Fréchet-Urysohn instead of ...
- 9,941
2
votes
1
answer
89
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Compact Hausdorff space, hereditarily Lindelöf but non-metrizable?
Is there a compact Hausdorff space that is hereditarily Lindelöf but non-metrizable? Under the continuum Hypothesis, such space exists (see the abstract of A compact L-space under CH by Kunen). I ...
- 1,308
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81
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Difference between a Partition and a Decomposition of a Set
I am reading an article by Erick K. van Douwen (The Integers and Topology) in which the author mentions the decomposition of a set without definition. I've found a definition that describes it as a ...
0
votes
1
answer
146
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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
1
vote
1
answer
70
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What's $(2^\mathfrak{c})^{<2^\mathfrak{c}}$?
I'm wondering if $\lambda = (2^{\mathfrak{c}})^{<2^\mathfrak{c}} = \sup_{\kappa < 2^\mathfrak{c}} 2^{\mathfrak{c}\cdot \kappa}$ can be calcuted without invoking the general continuum hypothesis ...
- 10.5k
2
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0
answers
52
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How many second-countable $T_1$ spaces are there? [duplicate]
How many second-countable $T_1$ spaces, up to homeomorphism, are there?
Let $X$ be a second-countable $T_1$ space. Since $X$ is second-countable, there are at most $\beth_1$ open sets in $X$. And ...
- 2,049
1
vote
0
answers
54
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Prove or disprove that if $f:X\longrightarrow Y$ is a perfect map then $w(Y)\le w(X)$
Let be $f$ a perfect map from a space $X$ to a space $Y$ so that I am trying to prove or disprove if the inequality
$$
\begin{equation}\tag{1}\label{1}w(Y)\le w(X)\end{equation}
$$
holds, where $w$ is ...
- 4,906
1
vote
1
answer
47
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Show that the Lindelof number of any space is less than or equal to the network weight
As a recap, the Lindelof number, $L(X)$, is the smallest infinite cardinal $\mathcal{K}$ so that every open cover of $X$ has a subcover with cardinality less than or equal to $\mathcal{K}$. Also the ...
- 2,782
0
votes
0
answers
19
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Number systems and Cardinality Question (to admit a bijection)
I am trying to complete the following proof:
Show that the real numbers are unique, in the sense that any complete
ordered field admits a bijection with R that preserves addition, multiplication, and ...
- 1
3
votes
4
answers
399
views
Connected set which is no-where path connected
Background: It's a fun exercise to try to construct a connected space $T$ such that no two points in $T$ can be connected with a path.
My solution to the puzzle was to use an order topology on a ...
- 6,340
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0
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121
views
(Generalized) Cantor set is uncountable
The standard Cantor set formed by recursively removing the middle one-thirds, on the interval $[0, 1]$ can be shown to be equal to the uncountable set of the base-3 numbers between $0$ and $1$ with ...
- 4,119
0
votes
1
answer
442
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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
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36
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Separating weight of discrete space.
Recall that a cover $\mathscr{A}$ of a set $E$ is separating if for each pair of distinct points $p,q\in E$ there is $A\in \mathscr{A}$ such that $p\in A$ and $q\not\in A$. The separating weight of a $...
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