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Results tagged with set-theory
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user 4354
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
27
votes
2
answers
2k
views
A set that can be covered by arbitrarily small intervals
Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of …
- 4,713
122
votes
Accepted
solving $f(f(x))=g(x)$
Q1: No. Let $g(0)=1, g(1)=0$ and $g(x)=x$ for all $x\in\mathbb R\setminus\{0,1\}$.
Assuming $f\circ f=g$, let $a=f(0)$, then $f(a)=1$ and $f(1)=g(a)=a$ since $a\notin\{0,1\}$.
Then $g(1)=f(f(1))=f(a)= …
- 4,432
20
votes
1
answer
907
views
A collection of intervals that can cover any measure zero set
This is a follow-up to this question (in fact, this is what originally motivated me to ask that one.)
Let's say that a sequence $\{s_i\}$ of positive reals covers a set $X\subset\mathbb R$ if there i …
CommunityBot
- 1
16
votes
Accepted
Improvements of the Baire Category Theorem under (not CH)?
A complete space without isolated points has at least continuum cardinality. At least if you agree to use (some form of) Axiom of Choice.
Choose two disjoint closed balls $B_1$ and $B_2$. Inside $B_1 …
- 32.4k
2
votes
Shape of long sequences in C(ω_1)
I can partially answer the second question. If $X$ is a compact Hausdorff space whose topology has a countable base at every point [Edit: $\omega_1$ has this property but not compact], then there are …
- 32.4k
22
votes
Set theory in practice
You don't have to define your objects as sets, in fact, you should avoid such unnatural definitions. I don't think a number theorist would be happy to see a proof referring to elements of a natural nu …
- 32.4k