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 Jan 1 revised The graph of a continuous function from a space $X$ into a Hausdorff space $Y$ is closed in $X \times Y$ added 6 characters in body; edited title Dec 31 revised Compact Hausdorff Spaces with pre-caliber $\aleph_1$ has caliber $\aleph_1$ phi is not the empty set, a is a weird variable for an ordinal Dec 31 revised What is a first countable, limit compact space that is not sequentially compact? added 87 characters in body Dec 31 revised What is a first countable, limit compact space that is not sequentially compact? added 62 characters in body Dec 30 comment What is a first countable, limit compact space that is not sequentially compact? @ZachStone This topology isn't too bad. It's sort of "half" the usual topology on $\mathbb {R}$, and can be used to characterise "lower semicontinuous functions". Dec 30 revised What is a first countable, limit compact space that is not sequentially compact? added 1263 characters in body Dec 30 revised What is a first countable, limit compact space that is not sequentially compact? added 89 characters in body Dec 30 answered What is a first countable, limit compact space that is not sequentially compact? Dec 29 comment Acceptable generic and non-circular definition of countable infinity? Comments are not for extended discussion; this conversation has been moved to chat. Dec 27 revised Gentle introduction to algebraic number theory link to pdf not exactly kosher Dec 26 comment The epsilon-delta definition of continuity @MathematicsStudent1122 Also functions like $f(x)= \frac{\sin (1/x)}{x}$ ($f(0)=0$) which takes the value $0=f(0)$ at points arbitrarily close to $x_0 = 0$. Dec 25 revised “Nested independence” of $\mathsf{ZFC}$? added 131 characters in body; edited tags; edited title Dec 21 comment Let $X$ an infinite $T_1$ space, then exist some subspace homeomorphic to $(\Bbb N,\tau)$ where $\tau$ is discrete or cofinite @DanielFischer Your comment is perhaps a bit pedantic. There is no claim that this is a Choiceless proof. Furthermore, if the exercise is to show that every infinite T₁ space has a subspace homeomoprhic to $\mathbb{N}$ with either the cofinite or discrete topology, then we have to assume that every infinite set has a countably infinite subset (i.e., there are no infinite Dedekind finite sets, which is somewhat weaker than the Axiom of Countable Choice). Dec 21 revised Compact sets in the lower topology on $\mathbb{R}$ have a minimum no more sequence. Dec 20 revised Compact sets in the lower topology on $\mathbb{R}$ have a minimum spelling things out in excruciating detail Dec 20 comment Is it consistent that every set is the countable union of sets with smaller cardinality, or is it just alephs? @Wojowu: Part of it states, $N_G \models \text{'every set is a countable union of sets of smaller cardinality'}$. Dec 20 comment Is it consistent that every set is the countable union of sets with smaller cardinality, or is it just alephs? Then go further to Theorem 6.3 (p.86). Dec 20 comment Is it consistent that every set is the countable union of sets with smaller cardinality, or is it just alephs? Look at the statement of Theorem II. Dec 20 revised Can a point and a compact set in a Tychonoff space be separated by a continuous function into an arbitrary finite dimension Lie group? deleted 96 characters in body; edited title Dec 20 comment $A$ be a subset of $[0,1]$ with non-empty interior ; then is it true that $\mathbb Q+A=\mathbb R$? I think you are confusing the sum $+$ (i.e., $A+B = \{ a+b : a \in A, b \in B \}$) with the union $\cup$.