16
votes
1answer
461 views

Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
5
votes
1answer
445 views

Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
6
votes
11answers
2k views

A function that is defined everywhere but has unknown values [closed]

For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...
1
vote
2answers
428 views

Uncountability of the “Peculiar” sets:

I call a set PECULIAR, if its elements are uncountable, pairwise disjoint subsets of R (the real number system). As for example, the set {(0,1),(3,5),[8,9]\Q},where Q denotes the set of rationals, is ...
1
vote
3answers
490 views

Ultrafilters and principal filters [closed]

Can someone give me an example of an ultrafilter which is not principal?
7
votes
2answers
2k views

Definition of Function

What is authoritative canonical formal definition of function? For example, According to Wolfram MathWorld, $$isafun_1(f)\;\leftrightarrow\; \forall a\in f\;(\exists x\exists y \;\langle x,y\rangle ...
3
votes
1answer
423 views

What are the oldest illustrations of “Venn” diagrams?

Graphical representations of intersection of sets as logical combinations are much older than Venn. Euler and Leibniz are often quoted and the current Wikipedia article also quotes Ramon Llull but I ...
15
votes
3answers
2k views

Model category structure on Set without axiom of choice

There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are ...
8
votes
5answers
924 views

Questions about ordering of reals and irrationals

Three problems from G.Rosenstein "Linear orderings" (from the end of Chapter 2 and beginning of Chapter 4): 1) Is there a nondecreasing function from irrationals onto reals? 2) Is there a ...