For questions about realcompact spaces. A Tychonoff space is realcompact if every real z-ultrafilter on it is fixed. Equivalently, a space is realcompact if it can be embedded as a closed subspace of $\mathbb{R}^\kappa$ for some cardinal $\kappa$. For reference see "Rings of continuous functions" by Gillman and Jerison.
Questions tagged [realcompact-spaces]
16 questions
9
votes
1 answer
Mysior plane is not realcompact
Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup \{(x+\sqrt{2}+y, -y) : 0 < y < 1/n\}.$$ The space $X$…
Jakobian
- 15,280
6
votes
1 answer
A metrizable space is realcompact iff it has non-measurable cardinality?
A space is realcompact if its a closed subspace of an arbitrary product of real lines, with product topology.
A cardinal $\kappa$ is called measurable if there exists a (countably additive) $\{0, 1\}$-valued measure $\mu:\kappa\to \{0, 1\}$ with…
Jakobian
- 15,280
5
votes
0 answers
Is Bing's discrete extension space realcompact?
See here for definition of Bing's space. There's also Dan Ma's blog or Counterexamples in Topology by Steen and Sebach.
Since the Michael's closed subspace $Y\subseteq X$ of Bing's space $X$ is metacompact but not paracompact, $Y$ is not countably…
Jakobian
- 15,280
4
votes
1 answer
Inductive limit of compact Hausdorff spaces being realcompact
Suppose that $\{X_n\}_{n\in \mathbb{N}}$ is an increasing sequence of compact Hausdorff spaces and $X$ is their union, equipped with the finest topology under which all the inclusion maps are continuous. Is it well-known that $X$ is realcompact? Is…
Chi-Keung Ng
- 41
4
votes
1 answer
Existence of non-discrete Tychonoff extremally disconnected P-space and measurable cardinals
Assume all spaces are Tychonoff.
In exercise $12$H of Rings of continuous functions by Gillman and Jerison they ask to prove the following two things:
If $X$ is an extremally disconnected $P$-space of cardinality smaller than first measurable…
Jakobian
- 15,280
4
votes
1 answer
Thomas plank is not realcompact
Let $X = \bigcup_{n\geq 0} L_n$ where $L_n = [0, 1)\times\{1/i\}$ for $i > 0$ and $L_0 = (0, 1)\times \{0\}$.
Define the topology on $X$ as follows: each point $(x, 1/i)$ for $x\in (0, 1)$ and $i > 0$ is isolated, neighbourhoods of $(0, 1/i)$ are…
Jakobian
- 15,280
4
votes
1 answer
Compactness, pseudocompactness, and realcompactness without Hausdorff
A space is compact provided every open cover admits a finite subcover.
A space is pseduocompact provided every continuous image of the space into the Euclidean line $\mathbb R$ is bounded.
A space is realcompact provided it embeds as a closed…
Steven Clontz
- 8,773
3
votes
1 answer
Is every metric space realcompact?
Let $X$ be a space and $\beta X$ denote the Stone-Cech compactification of $X$.
$X$ is realcompact if for each $p\in \beta X\setminus X$ there is a continuous $f:X\to[0,\infty)$ such that $\beta f (p)=\infty$, where $\beta f:\beta X\to[0,\infty]$…
Forever Mozart
- 9,534
2
votes
1 answer
Does a realcompact space have unique realcompactification it is $G_\delta$-dense in?
In literature there exists the following definition:
$Y$ is a realcompactification of $X$ if $X$ is dense in $Y$ and $Y$ is realcompact.
What I actually want to define instead is this:
$Y$ is a realcompactification* of $X$ if $X$ is…
Jakobian
- 15,280
1
vote
1 answer
Realcompactification of Rudin's Dowker space
Let $$X = \{f\in \prod_n (\omega_n+1) : \exists_{i} \forall_n\omega_0 < \text{cf}(f(n)) < \omega_i\}$$ be Rudin's Dowker space and $$X' = \{f\in \prod_n (\omega_n+1) : \forall_n \omega_0 < \text{cf}(f(n))\}.$$
In the original paper by Rudin, namely…
Jakobian
- 15,280
1
vote
1 answer
When is a disjoint union of topological spaces realcompact?
Let $X = \bigsqcup_{i\in I}X_i$ be a disjoint union of non-empty topological spaces $X_i$.
Is there any condition on spaces $X_i$ which would imply that $X$ is realcompact?
A Tychonoff space is realcompact when it can be embedded as a closed…
Jakobian
- 15,280
1
vote
2 answers
What realcompact spaces fail to be Lindelöf?
I'm working on this pull request for the pi-Base and need to add peer reviewed counterexamples to the theorem that all $T_3$ Lindelöf spaces are realcompact, that is, a closed subset of some (not necessarily finite) power of the Euclidean line…
Steven Clontz
- 8,773
1
vote
1 answer
$X$ is compact Hausdorff iff it is pseudocompact and realcompact
I just read this article http://en.wikipedia.org/wiki/Realcompact_space. I am interested with a property: $X$ is compact Hausdorff iff it is pseudocompact and realcompact.
I don't know how to prove this property. Since have not this book…
flourence
- 429
0
votes
1 answer
Is this an open cover? (From Thm: Distribution function implies unique probability)
I am reading both the proof and the lecture notes of the theorem and I don't understand the following. The proof is in a 1996 book "Probability" by Shiryaev (ISBN 978-1-4757-2541-4).
We have $(A_n)_{n=1}^\infty$ a decreasing sequence of sets,…
Waaal
- 377
- 2
- 11
0
votes
2 answers
Do Continuous Functions Preserve Realcompactness?
My definition for Realcompact Space: $X$ is realcompact if it can be embedded as a closed subspace of a product of copies of the real line.
Do continuous functions preserve realcompactness?
To me, it seems likely, as many other notions of…
Ishan Deo
- 3,886