Linked Questions

2
votes
1answer
1k views

Baire's Theorem and Irrationals [duplicate]

I am asked to show that the irrational numbers are not a countable union of closed subsets of $\mathbb{R}$ given that if a complete metric space is the countable union of of closed subsets then at ...
2
votes
0answers
218 views

Why is $\Bbb Q$ not $G_{\delta}$? [duplicate]

I know of proofs that the rationals are not $G_{\delta}$, so I was just wondering what the following set is equal to: $$ \bigcap_m \bigcup_{r_n\in \Bbb Q}(r_n-2^{-n}/m,r_n+2^{-n}/m) $$ I know it ...
0
votes
0answers
58 views

prove that $\mathbb{Q}$ can not be $G_{\delta}$ set in $\mathbb{R}$. [duplicate]

prove that $\mathbb{Q}$ can not be $G_{\delta}$ set in $\mathbb{R}$. let $\mathbb{Q}$ is $G_{\delta}$ set. then $\mathbb{Q}=\cap_{i=1}^{\infty}\ O_i$. for open sets $O_i$. now each open set in $\...
1
vote
0answers
44 views

$\mathbb{Q}$ is not the countable intersection of open sets. Is my Reasoning Correct? [duplicate]

The question has actually 2 parts. I know the question has been asked before, but I want to know if my reasoning makes sense.I've done the first part but am unsure about the second. It goes as follows:...
46
votes
1answer
9k views

Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$

I'm looking for subset $A$ of $\mathbb R$ such that $A$ is a Borel set but $A$ is neither $F_\sigma$ nor $G_\delta$.
3
votes
2answers
2k views

Example of non G-delta set

An open set is clearly a $G_{\delta}$ set. A closed interval $[a,b]$ is a $G_{\delta}$ set as an intersection of the open intervals $(a-\frac1n,b+\frac1n)$ for all positive integers $n$. What is an ...
7
votes
2answers
2k views

$G_\delta$ sets

I understand that a $G_\delta$ set is a set which is a countable intersection of open sets. My question is: Is there any other characterization for $G_\delta$ sets (on $\mathbb{R}$)? For example, can ...
2
votes
2answers
135 views

$\mathbb{Q}$ is not a $G_{\delta}$ What is wrong with my argument?

I am trying to show that $\mathbb{Q}$ is not a not a $G_{\delta}$. However, I am confused with the following argument: Clear for all $r \in \mathbb{Q}$, $(r-\frac{1}{n} ,r+\frac{1}{n})$ is an open ...
5
votes
1answer
244 views

Would this space be homeomorphic to the set of irrationals?

I've been reviewing various problems dealing with interesting homeomorphisms, and I came across this one. Is the product of the space of irrationals and the space of rationals homeomorphic to the ...
5
votes
3answers
220 views

$f: \mathbb{Q} \rightarrow \mathbb{R} \ \ \lim _{q \rightarrow t, \ q\in \mathbb{Q}} f(q) =g$

There's a continuous function $f: \mathbb{Q} \rightarrow \mathbb{R}$ Prove that $ \exists t \in \mathbb{R} \setminus \mathbb{Q} \ \ \exists g \in \mathbb{R} :\ \ \lim _{q \rightarrow t, \ q\in \...
4
votes
1answer
165 views

Is it possible to have $D=\Bbb P$

Let $f:\Bbb R\to \Bbb R$ and $D=\{x\in \Bbb R: f $ is discontinuous at $x\}$. My problem is : Is it possible to have $D=\Bbb P$ where $\Bbb P$ is the set of irrationals in $\Bbb R$. I know the ...
1
vote
2answers
122 views

Problem on Baire's category theorem.

I'd like to show that the set of irrational numbers in $[0,1]$ cannot be represented as a countable union of closed sets. The hint says to use Baire's category theorem. I know two versions of such ...
2
votes
2answers
144 views

Is every generalized-$F_{\sigma}$ set an $F_{\sigma}$ set?

A subset $S$ of a topological space $X$ is called a generalized-$F_{\sigma}$ set in $X$ if for all open $G \subset X$ with $S\subset G$, there exists an $F_{\sigma}$-set $F$ such that $S\subset F\...
2
votes
0answers
282 views

$\mathbb{R}\setminus \mathbb{Q}$ is not an $F_\sigma$

I have a topology question regarding the proof that $\mathbb{R}\setminus \mathbb{Q}$ is not an $F_\sigma$. The proof is very informal and would like to receive some formal explanation because I could ...
0
votes
1answer
89 views

Rationals can be the set of continuity of a function? [duplicate]

Most of the functions that I have seen have their discontinuities on rationals and continuities on irrationals! I am wondering if there is any exampe of some function whose continuities are rationals?...

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