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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 $X$.)
Feb
8
comment Size of the closure of a set
Yes. It's a great book if you're studying set-theoretic topology.
Feb
8
awarded  Editor
Feb
8
comment Size of the closure of a set
Thanks, I stuck in the forcing construction, since it's not a priori obvious how to force such a model.
Feb
8
revised Size of the closure of a set
added 613 characters in body
Feb
8
awarded  Revival
Feb
8
answered Size of the closure of a set
Feb
8
awarded  Supporter
Feb
8
comment Topology and axiom of choice
Thanks Andres! "General topology under the axiom of determinacy" can be found at www(dot)math(dot)berkeley(dot)edu/(tilde)apollo/gen_top_ad.ps.gz for those curious.
Feb
8
answered Topology and axiom of choice
Feb
8
answered Order relation between Ordinals
Feb
4
awarded  Teacher
Feb
4
answered Striking applications of integration by parts