22 questions linked to/from Perfect set without rationals
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Closed, uncountable set without rationals. [duplicate]

Problem: Find a closed, uncountable without rationals, subset in the unit interval. Solution: Let X be any binary irrational number , replace the 1s with 2s in the decimal expansion . Now we can ...
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Perfect set in $\mathbb{R}$ which contains no rational number [duplicate]

Possible Duplicate: Perfect set without rationals Does there exist a nonempty perfect set in $\mathbb{R}$ which contains no rational number? This problem is on p.44 PMA - Rudin I found a proof ...
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Perfect set without rational numbers [duplicate]

Sorry if this problem is repeated. Is there a nonempty perfect set in $\mathbb{R}^1$ which contains no rational number? Proof sketch: This set must be uncountable because any nonempty perfect set in ...
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Rudin exercise 2.18: perfect set with only irrational numbers. [duplicate]

I am trying to solve exercise number 2.18 in Rudin. The question is the following: "Is there a nonempty perfect set in R1 that contains no rational numbers"? My solution is basically to construct a ...
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Rudin's PMA Exercise 2.18 - Perfect Sets [duplicate]

I've been working through Chapter 2 questions and have thought about Exercise 2.18 for a while, but couldn't come up with an answer. Is there a nonempty perfect set in R which contains no rational ...
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Is there a non-empty subset of $\mathbb{R}$ like $A$ such that the set of accumulation points of $A$ is itself and $A\cap\mathbb{Q} = \emptyset$ [duplicate]

Is there a non-empty subset of $\mathbb{R}$ like $A$ such that the set of accumulation points of $A$ is $A$ and $A\cap\mathbb{Q}=\emptyset\,$?
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I encountered this question in Abbott's Understanding Analysis. The problem asks to construct a nonempty perfect set with no rationals. It starts with enumerating the rationals $\mathbb{Q}=\{r_1,r_2,..... 0answers 16 views Basic Topology about metric space$R$[duplicate] Is there a nonempty perfect set in$R$which contains no rational number? Its the excercise of Principles of Mathematical Analysis by Walter Rudin. I can't find the answer.I try to construct a set as ... 3answers 2k views Irrational Cantor set? Can someone help me? How can I prove that exists a number$k \in \mathbb R$that $$A = \{x + k;\ x \in \text{Cantor set} \} \subset\text{ Irrationals}\;?$$ 4answers 708 views Questions about open sets in${\mathbb R}$Consider the following problem: Let${\mathbb Q} \subset A\subset {\mathbb R}$, which of the following must be true? A. If$A$is open, then$A={\mathbb R}$B. If$A$is closed, then$A={\mathbb R}$... 1answer 2k views Is there a compact subset of the irrationals with positive Lebesgue measure? Does there exist$K \subseteq \mathbb{R} \backslash \mathbb{Q}$such that$K$is compact, and has Lebesgue measure greater than$0$? As I have been trying to think of examples, I suspect that any ... 2answers 283 views Is it possible that all subseries converge to irrationals? Does there exists a positive decreasing sequence$\{a_i\}$with$\sum_{i\in\mathbb{N}} a_i$convergent, such that$\forall I\subset\mathbb{N},\sum_{i\in I}a_i$is an irrational number? Such an ... 4answers 856 views Discontinuous function at an uncountable set with not rationals Does there exists a function$f:[0,1]\to\mathbb{R}$such that$D(f)$(its points of discontinuity) is an uncountable set containing no rational number? First thing I thought of was$\mathbb{R}\...
How to show : If $A \subseteq \mathbb{I}$ is not countable, then $A$ is not closed set in $\mathbb{R}$. I think that if $A$ is closed, then $A^{c}$ is open thus: $A^{c}$ is an union of open ...