I'm reading "Contemporary Abstract Algebra," by Gallian.
This is (part of) Exercise 26 of the supplementary exercises for chapters 1-4 ibid., although I am requesting a proof of the contrapositive just out of interest, preferably using prior material available from the textbook.
I intend to use...
Hey everyone! Suppose we have an union of two countable sets, $A$ and $B$. We can enumerate their union like this: $ \{ b_1, a_1, b_2, a_2, b_3, a_3 ... \}$.
Now, let's define a function $f : S \rightarrow \mathbb{N} $, where S is the union of A and B, defined as $f(s_n) = 2n$, if $s_n \in A$, and $f(s_n) = 2n-1$, if $s_n \in B$. Do you think this is a valid bijection from S to N?
For example, if the elements of the union are b1, a1, b2, a2 etc, we have 1, 2, 3, 4...
You can do something similar for paracompact subspaces
and when a paracompact subspace is embedded in this nice way, that you can apply definition of paracompactness on it from within the whole space, those are called strongly paracompact subspaces
And for $T_4$ spaces we can characterize those as $P$-embedded paracompact subspaces
I found it because I thought one question was asking about equivalence of paracompactness and strong paracompactness for closed subspaces, but they might have just asked if closed subspace of paracompact space is strongly paracompact
because I thought that when they mention closed subspaces of paracompact space are paracompact, that was just an example to get you interested in the question
This is added by lichess on top of stockfish's analysis
Probably the game review, which has some fixed depth, marked it as inaccurate, but if you let the engine sit on it for a while it evaluates it differently
You can easily take this limit involving twin primes, but perhaps that's not true: https://math.stackexchange.com/questions/4718413/the-asymptotic-limit-of-counting-twin-primes-in-p-n-2-p-n2-would-be-easi
What I mean, is if the rules were lemmatized out right, you could do it
$$\lim_{n \to \infty} \frac{1}{p_{n}^2} \sum_{d \mid p_n\#}(-1)^{\omega(d)}\sum_{r^2 = 1 \pmod d}\left\lfloor \frac{p_n^2 - r}{d}\right\rfloor$$ exists and is $\gt 0$ is the Twin Prime Conjecture._ Idk if you knew that @copper.hat
Or it might be stronger, anyway, it's sufficient. Isn't it a beautiful formula?
@copper.hat yes, I know but I've never seen this limit taken before. And I believe it is the "correct" one in a sense that it will lead to a nice elegant, elementary proof.
It involves Mertens estimate however, which is not elementary
@TedShifrin whenever you return, I'm doing 6.1.7 for a bit of extra practice with getting estimates. I'm having trouble trying to figure out the $\delta$ to obtain continuity. I have seen something online and it is reasonable, but it doesn't make sense to me how they could arrive at that based on the prior parts of the question
@THE_CRANIUM depends for what, the explicit formula is a recursive function of the set of strings generated by 1,0
if you want to prove it is a bijection you need to make it explicit, otherwise you should just convince yourself informally that that's true
i just realized, you can give a proof without fully formalizing everything, injectivity can be proven by induction and surjectivity can be proven directly, assuming nobody asks you what the symbols mean heheh
the inductive proof for injectivity is the same as the one i told you earlier today
