# Tagged Questions

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### “Partitioning” an uncountable set “equally”

I just observed a simple fact about the real numbers, that if U is an uncountable subset of the set of real numbers, then there exists a real number r, such that both the set V of all elements of U ...
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### Flaw in the proof that a set is countable [duplicate]

Q: Let S be the set containing all sequences of 0's and 1's. i.e $S = \{(a_1,a_2,a_3,a_4,\ldots) : a_i = 0 \text{ or } 1\}$ Show that S is countable. Proof(Flawed) : Let $A_i$ be the ...
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### Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
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### Theorem of Galvin, Mycielski and Solovay

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of ...
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### An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an "ordered and infinite list of real ...
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### When the continuum hypothesis settles the uniqueness (upto isomorphism) of the Hyper-reals doesn't it mean the hypothesis should be an axiom?

One useful consequence of $ZFC$ is that the real numbers can be shown to be unique upto isomorphism. According to wikipedia: The use of the definite article the in the phrase the hyperreal ...
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### Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?

Is there a simple, constructive, 1-1 mapping between the reals and the irrationals? I know that the Cantor–Bernstein–Schroeder theorem implies the existence of a 1-1 mapping between the reals and the ...
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### Cardinality of set of discontinuities of cadlag functions

I know that non-decreasing cadlag functions (functions that are right continuous with left limits) on $[0,\infty)$ have at most a countable number of discontinuities. Does the same result hold for ...
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### Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?

A UC Berkeley prelim exam problem asked whether an additive function $f\colon {\mathbb R} \to {\mathbb R}$, i.e. satisfying $f(x + y) = f(x) + f(y)$ must be continuous. The counterexample involved ...
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### Can we find a family of sets over the whole space $X$ that is “larger” than the power set of $X$

Don't know if the statement in the title is true or not. Can anyone help me out? Thanks!
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### How is the extended real number line modeled?

In set theory, numbers are often constructed, e.g. from nestings of sets which eventually contain the empty set. The operations are defined in term of taking unions etc.etc. The extended real number ...
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### Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?

I recently started studying set theory and am having quite a bit of difficulty accepting Cantor's diagonal proof for the uncountability of the reals. I also saw a topological proof via nested sets for ...
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### how to create the set of hyperreal numbers using ultraproduct

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?
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### Are there any continuous functions from the real line onto the complex plane?

Is there any measurable continuous differentiable analytic surjective function $f:\mathbb{R}\to\mathbb{C}$?
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### The set S of all alignments of digits obtainable by sequence of positive integers is countable?

Is $\left\{ n_k \right\}_{k=1}^{\infty}$ a strictly increasing sequence of positive integers (written in decimal notation). Consider the alignment (infinite) set of digits {${n_1 n_2 n_3}$} format ...
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### Questions about $\sigma$-algebra, algebra and topology

I know the definitions of $\sigma$-algebra, algebra and topology, but why countable/finite union, as in $\sigma$-algebra/algebra, and finite intersection, arbitrary union as in topology? What inspire ...
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### Is this function constructed using AC necessarily discontinuous everywhere?

Assume AC. Let $x_\alpha$ be a well-ordering of $\mathbb{R}$. For all $\alpha < \mathfrak{c}$, let $F(x_\alpha) = x_{\alpha+1}$. Can it be proven that $F$ is discontinuous everywhere?
I know that the set of continuous functions on $R^n$ has cardinality of R. Can this be generalized to any subspace of $R^n$? It seems intuitive, but the empty set seems to be a counterexample of this ...