Reputation
8,379
Top tag
Next privilege 10,000 Rep.
Access moderator tools
Badges
1 22 51
Newest
 Enlightened
Impact
~121k people reached

Apr
23
comment “Downward closed” relation on a poset
Downward closed sets are sometimes called "order ideals".
Apr
17
awarded  Enlightened
Apr
17
comment Cardinality of connected Hausdorff topologies
My answer suggests that a more interesting question might have been "how many locally connected Hausdorff spaces...". But I am pretty sure that the answer would stay the same.
Apr
17
comment Cardinality of connected Hausdorff topologies
Call the original space $A$, the "strings" (copies of the open unit interval or connected space) $B$, and let $c$ be the point where are strings are glued together. As Jeremy Rickard points out, the very many topologies on $A$ induce very many topologies on $A\cup B\cup \{c\}$, and the number of homeomorphism classes stays "very large" (i.e., $2^{2^\kappa}$). -- However, at least in the case $\kappa\ge 2^{\aleph_0}$, it is clear that each autohomeomorphism must map $B$ onto $B$ (all neighborhoods are $\simeq (0,1)$), and $c$ to $c$, hence $A$ onto $A$.
Apr
17
awarded  Nice Answer
Apr
17
revised Cardinality of connected Hausdorff topologies
deleted 12 characters in body
Apr
17
answered Cardinality of connected Hausdorff topologies
Apr
8
comment $\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?
Welcome to math overflow! Nice to see you here!
Mar
22
comment Reference request, proof about well-ordering of ordinals
"A system where you can quantify over predicates" describes the language of second order logic. This description does not tell you which proofs you allow. One way (the cheap way, but still good enough for most of mathematics) is to translate second order statements into first order statements in the language of ZFC, and then use your favorite first order proof system (plus ZFC axioms, or more) to analyse these statements.
Mar
21
revised A weak kind of fixed point
tag: lattice-theory
Mar
21
comment Is there a particular field that cannot be proven to have an algebraic closure in ZF?
(A generalisation of @JulianRosen's comment:) For any ordered (i.e., linearly ordered) field $K$, the ordered finite extensions form a directed system. (The embeddings between them are canonical, because of the order.) Their naturally defined limit (without AC, I think) is a real closed field. Adjoin an imaginary unit $i$ to get an algebraically closed field.
Mar
20
comment Reference request, proof about well-ordering of ordinals
There is no canonical notion of "second order proof". Which notion do you have in mind?
Mar
2
answered Why should we believe in the axiom of regularity?
Feb
22
revised Bidi: A new cardinal characteristic of the continuum?
polished Polish
Feb
17
comment When can you canonically extend an ultrafilter after forcing?
Do you have an example of a forcing that has this property?
Feb
15
comment Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?
The title of your question is misleading, because the structure $(\{1,2,\ldots\,\},\oplus)$ is isomorphic to the well-known structure $(\{0,1,2,\ldots\, \},+)$. You are interested in $\{1,2,\ldots\},\oplus,\cdot)$.
Jan
15
answered Is there an uncountable Borel almost disjoint family?
Dec
29
comment Unconventional types of induction
Thanks for fixing it.
Dec
17
revised Are undecidable consequences of Con recursively enumerable?
\\, -> \, (TeX)
Dec
11
comment Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?
If you drop the separability assumption, a similar fact is true: An infinite-dimensional vector space over the reals of dimension $\kappa$ admits a complete norm if and only if $\kappa = \kappa^{\aleph_0}$. (Arthur Kruse, Badly incomplete normed linear spaces. Mathematische Zeitschrift, 1964.)