All Questions
Tagged with model-theory gn.general-topology
23
questions
4
votes
0
answers
281
views
Which countable sets don't drastically change the definable topologies on $\mathbb{R}$?
For $\mathcal{M}$ an expansion of $\mathcal{R}=(\mathbb{R};+,\times)$ and $A\subseteq\mathbb{R}$, let $\tau^\mathcal{M}_A$ be the topology on $\mathbb{R}$ generated by the sets definable in $\mathcal{...
- 20.7k
11
votes
1
answer
665
views
On the classification of second-countable Stone spaces
Let $X$ be a Stone space (i.e. totally disconnected compact Hausdorff). Then the following are equivalent:
$X$ is second countable
$X$ is metrizable
$X$ has countably many clopen subsets
$X$ is an ...
- 62.6k
14
votes
0
answers
419
views
Which functions have all the common $\forall\exists$-properties of continuous functions?
This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well.
For a ...
- 20.7k
8
votes
0
answers
239
views
First order formula describing connected components
I ask this question here after no answer came up in the original MathSE question.
Let $\mathcal{L}$ be the language $\{+,-,\cdot,0,1,P\}$ where $P$ is some $n$-ary relation symbol. Is there a formula $...
- 458
19
votes
0
answers
555
views
What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?
Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
- 20.7k
9
votes
0
answers
256
views
Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?
Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
- 20.7k
10
votes
0
answers
324
views
Extending models of topological set theory
$\mathsf{GPK_\infty^+}$ is an alternative set theory in which we have comprehension for formulas which are positive in a certain sense; see the SEP article for more detail (or this MO post, which ...
- 20.7k
18
votes
1
answer
1k
views
A topological version of the Lowenheim-Skolem number
This is a continuation of an MSE question which received a partial answer (see below).
Given a topological space $\mathcal{X}$, let $C(\mathcal{X})$ be the ring of real-valued continuous functions on $...
- 20.7k
9
votes
1
answer
403
views
On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces
In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are ...
- 3,011
0
votes
2
answers
213
views
Intrinsically defining smooth/continuous/analytic functions
In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...
user158636
12
votes
0
answers
240
views
Is there a characterization of the class of first-order formulas that are closed in every compact Hausdorff structure?
Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ ...
- 10.4k
10
votes
1
answer
348
views
Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?
Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures.
Consider the endomorphism $\hat{\Phi}$ ...
- 62.3k
1
vote
1
answer
999
views
A new generalisation of dimension? part 2
I worked this theory : A new generalization of the dimension?
I have a theorem about dimensions which is more general and simple than for matroids.
Definition 1: A structure $S$, is a pair $(X, \...
- 3,824
0
votes
0
answers
640
views
A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 3,824
19
votes
0
answers
924
views
What is the Cantor-Bendixson rank of the space of first order theories?
Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
- 2,136