### A constructive proof of the Tychonoff's theorem for locales

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We prove that the Baire Category Theorem is equivalent to the following: Let G be a topological groupoid such that the unit space is a complete metric space, and there is a countable cover of G by neighbourhood bisections. If G is effective, then G is topologically principal.

We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson: "If a normed space E does not contain any asymptotically isometric copy of l1, then every bounded sequence of E' has a normalized l1-block sequence pointwise converging to 0".

Given a set $X$ of “indeterminates” and a field $F$, an ideal in the polynomial ring $R=F\left[X\right]$ is called conservative if it contains with any polynomial all of its monomials. The map $S\mapsto RS$ yields an isomorphism between the power set $\text{P}\phantom{\rule{0.166667em}{0ex}}\left(X\right)$ and the complete lattice of all conservative prime ideals of $R$. Moreover, the members of any system $\text{S}\phantom{\rule{0.166667em}{0ex}}\subseteq \text{P}\phantom{\rule{0.166667em}{0ex}}\left(X\right)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in ${P}_{\text{S}}=\bigcup \{RS:S\in \text{S}\phantom{\rule{0.166667em}{0ex}}\}$, and the maximal members of $\text{S}\phantom{\rule{0.166667em}{0ex}}$ correspond to the maximal ideals contained in...

We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.

We show that AC is equivalent to the assertion that every compact completely regular topology can be extended to a compact Tychonoff topology.

This paper extends the notion of an archimedean frame to frames which are not necessarily algebraic. The new notion is called joinfitness and is Choice-free. Assuming the Axiom of Choice and for compact normal algebraic frames, the new and the old coincide. There is a subfunctor from the category of compact normal frames with skeletal maps with joinfit values, which is almost a coreflection. Conditions making it so are briefly discussed. The concept of an infinitesimal element arises naturally,...