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Feb
11
revised Cardinal arithmetic: $b\ge x>1$ and $b^2=b$ implies $x^b=2^b$
added 814 characters in body
Feb
11
answered Cardinal arithmetic: $b\ge x>1$ and $b^2=b$ implies $x^b=2^b$
Feb
11
revised Cardinal arithmetic: $b\ge x>1$ and $b^2=b$ implies $x^b=2^b$
Texified and corrected problem with $x=1$.
Feb
11
suggested approved edit on Cardinal arithmetic: $b\ge x>1$ and $b^2=b$ implies $x^b=2^b$
Feb
11
answered What is an efficient algorithm to compute modular exponentiation of stacked exponents?
Feb
11
answered Fastest way to compute HCF of 2 numbers
Feb
10
comment Cardinality of all lines on $\mathbb R^{2}$ which do not contain point $(x,y)\in l$ where $x, y \in \mathbb Q$
How many irrationals are there? (If there are $\aleph_0$ rationals and $2^{\aleph_0}$ total real numbers, then ...) There is one horizontal line for each irrational. This will give you a lower bound (since there are lots of non-horizontal lines that miss rational points) on the number of lines you are looking for.
Feb
10
comment Stone-Cech compactification problem
Yeah, and for general spaces, i guess the same argument should work (if one is okay with non-Hausdorff compactification.)
Feb
10
answered Injective maps from $B^A$ to $(C^B)^{C^A}$
Feb
10
answered Cardinality of all lines on $\mathbb R^{2}$ which do not contain point $(x,y)\in l$ where $x, y \in \mathbb Q$
Feb
10
answered Stone-Cech compactification problem
Feb
9
awarded  Revival
Feb
9
answered Alexandroff compactification question
Feb
9
awarded  Mortarboard
Feb
8
revised Cardinality of $P(\mathbb{R})$ and $P(P(\mathbb{R}))$
added 126 characters in body
Feb
8
answered Cardinality of $P(\mathbb{R})$ and $P(P(\mathbb{R}))$
Feb
8
awarded  Necromancer
Feb
8
comment Size of the closure of a set
Yes, I wouldn't be surprised if there weren't an independence result here, something like $\text{MA}+2^{\aleph_0}=\aleph_2$ implying that every separable, sequentially-compact, (+compact?) space is $\leq\aleph_2$. For example, there's the result that if $\mathfrak{c}\leq\aleph_2$ then every compact, sequentially-compact space is pseudo-radial, etc.
Feb
8
comment Size of the closure of a set
One more comment: the claim at that "For pseudoradial Hausdorff spaces X we have |X| <= d(X)^c(X) (<= 2^d(X))" (at the at.yorku.ca link) is not correct (it is the second inequality that is a problem.) Otherwise we would have a fine contradiction coming from the fact that if $\mathfrak{c}\leq\aleph_2$ then all compact, sequentially compact spaces are pseudo-radial, but my (consistent) counterexample is a separable, compact, sequentially-compact space (hence pseudo-radial) and $\mathfrak{c}=\aleph_2$ but $|X|>2^{\aleph_0}$.
Feb
8
comment Size of the closure of a set
This is the Hewitt-Marczewski-Pondiczery Theorem: let $\kappa\geq\aleph_0$ be a cardinal and $\beta\leq2^\kappa$. Then if $X_\alpha$ are topological spaces with $d(X_\alpha)\leq\kappa$ for $\alpha<\beta$ then $d(\prod_{\alpha<\beta}X_\alpha)\leq\kappa$. In our case, we are using the special case for $\kappa=\aleph_0$: the product of no more than continuum many separable spaces is separable. We are taking the product of $\omega_1<2^\omega$ many finite spaces. ($d(X)$ is the density of a space $X$.)