QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spa …

Edit: Zhen Lin incisively observes in a comment below that the category of compact Hausdorff spaces is monadic over the category of sets, hence is cocomplete. That answers the firs …

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spa …

I am just curious: is there a published proof of the compactness of the Hilbert cube that does not use the Axiom of Choice, or is it well known?

Hi, doing my research I found the following problem and I´ll be glad if someone could give a reference. We say that a compact connected subset $K$ of the plane is psuedo lamina …

At http://en.wikibooks.org/wiki/Real_Analysis/Metric_Spaces you can find the standard definition of a metric space: a set $X$ given with a function $d:X\times X\to\mathbb{R}$ that …

Consider the following subsets of $\mathbb{C}^n$ given by $$ X := {x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 } $$ $$ Y := { x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 } …

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $ …

Let $X$ be a closed subset of $\mathbb{R}^2$. What restrictions are there on $\pi_1(X)$ and on the homology groups of $X$ (both singular and Cech)? This is elementary if $X$ has …

In an attempt to write a proof by contradiction, I end up with a space $X$ with the following properties: (0) $X$ is nonempty, (1) $X$ is Hausdorff, (2) $X$ has no isolated points …

Let $\omega$ be a nonprincipal ultrafilter on $\mathbb N$. A standard construction gives an $\omega$-limit, say $x_\omega$, for any bounded sequence $(x_n)$ of real numbers. Name …

(Or: "I heard you liked Cantor Sets...") I'm working on a student project, and the following construction came up very naturally: If $C$ is the usual Cantor Set, build a countable …

Let $f$ be a $C^\infty$ function on $(c,d)$ ,and let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$. Su …

Let $(X,\tau)$ be a Tychonoff Topological space. For each $x\in X$ consider an arbitrary positive real number $\epsilon_x>0$. Is There a continuous real valued function $f:X\ri …

We Know that all closed subsets of a compact topological space $(X,\tau)$ are compact. But if we add The Hausdorff condition on the topology $\tau$ we could see the equivalence of …