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The topological space $$ X = \{ (x,\sin \left( \frac{\pi}{1-x} \right)) \ | \ 0 \leq x < 1 \} \cup {(1,1)} \subseteq \mathbb{R}^2 $$ has the FPP but its square does not. See Example $2$ p.$977$ of E. …

One can prove the following : (1) If $X$ has enough curves, then $X$ is sequential and locally path-connected. (2) If $X$ is first-countable and locally path-connected, then $X$ has enough curves. …

Is there some minimal idempotent ultrafilter $q \in \beta( \mathbb{N}^2)$ (with respect to the law $"+"$) such that any $A \in q$ is a subset of $\mathbb{N} \times \{ 0 \} $ ? (See for example http: …

As indicated by Theo Buehler above, Fremlin proves what I want and much more in his book. However, the proof given in this reference can be simplified a lot in my setting, so that I can answer my own …

Recently a student asked me the following (elementary looking) question : If $T$ is an invertible linear transformation of some finite-dimensional space $E$ into itself which factorizes as $T = f \ci …

Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system, with $T$ invertible, where the $\sigma$-algebra $\mathcal{B}$ is a Borel algebra arising from a topology which makes $T$ continuous, and …

Let $K$ be an object of $\mathbf{CHaus}$, the category of compact Hausdorff spaces, and $K \xrightarrow{\ \ \sigma \ \ } K$ be an involutory morphism without fixed points. Define $C^{\sigma}(K)$ as t …

An object $X$ of a given category is called projective if for each morphism $f : X \rightarrow Z$, and each epimorphism $ g : Y \twoheadrightarrow Z$, there is a morphism $h : X \rightarrow Y$ such th …

Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of non …