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 Feb11 revised Cardinal arithmetic added 814 characters in body Feb11 answered Cardinal arithmetic Feb11 revised Cardinal arithmetic Texified and corrected problem with $x=1$. Feb11 suggested approved edit on Cardinal arithmetic Feb11 answered What is an efficient algorithm to compute modular exponentiation of stacked exponents? Feb11 answered Fastest way to compute HCF of 2 numbers Feb10 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. Feb10 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.) Feb10 answered Injective maps from $B^A$ to $(C^B)^{C^A}$ Feb10 answered Cardinality of all lines on $\mathbb R^{2}$ which do not contain point $(x,y)\in l$ where $x, y \in \mathbb Q$ Feb10 answered Stone-Cech compactification problem Feb9 awarded Revival Feb9 answered Alexandroff compactification question Feb9 awarded Mortarboard Feb8 revised Cardinality of $P(\mathbb{R})$ and $P(P(\mathbb{R}))$ added 126 characters in body Feb8 answered Cardinality of $P(\mathbb{R})$ and $P(P(\mathbb{R}))$ Feb8 awarded Necromancer Feb8 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. Feb8 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 $|X|>2^{\aleph_0}$. Feb8 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 $X$.)