Choice principles in elementary topology and analysis

summary:Many fundamental mathematical results fail in {\bf{ZF}}, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results --- old and new --- that specify how much ``choice'' is needed {\it precisely} to validate each of certain basic analytical and topological results

E\mathbf{E}-compact extensions in the absence of the Axiom of Choice

The main aim of this work is to show, in the absence of the Axiom of Choice, fundamental results on $\mathbf{E}$-compact extensions of $\mathbf{E}$-completely regular spaces, in particular, on Hewitt realcompactifications and Banaschewski compactifications. Some original results concern a special subring of the ring of all continuous real functions on a given zero-dimensional $T_1$-space. New facts about $P$-spaces, Baire topologies and $G_{\delta}$-topologies are also shown. Not all statements investigated here have proofs in $\mathbf{ZF}$. Some statements are shown equivalent to the Boolean Prime ideal Theorem, some are consequences of the Axiom of Countable Multiple Choices

Idempotent ultrafilters without Zorn’s Lemma

We introduce the notion of additive filter and present a new proof of the existence of idempotent ultrafilters on N without using Zorn’s Lemma in its entire power, and where one only assumes the Ultrafilter Theorem for the continuum

An Introduction to Set Theory and Topology

These notes are an introduction to set theory and topology. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St. Louis. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and engineering. The usual background for the material is an introductory undergraduate analysis course, mostly because it provides a solid introduction to Euclidean space Rn and practice with rigorous arguments — in particular, about continuity. Strictly speaking, however, the material is mostly self-contained. Examples are taken now and then from analysis, but they are not logically necessary for the development of the material. The only real prerequisite is the level of mathematical interest, maturity and patience needed to handle abstract ideas and to read and write careful proofs. A few very capable students have taken this course before introductory analysis (even, rarely, outstanding university freshmen) and invariably they have commented later on how material eased their way into analysis.https://openscholarship.wustl.edu/books/1020/thumbnail.jp

Realcompact Alexandroff spaces and regular σ-frames

Bibliography: pages 96-103.In the early 1940's, A.D. Alexandroff [1940), [1941) and [1943] introduced a concept of space, more general than topological space, in order to obtain a simple connection between a space and the system of real-valued functions defined on it. Such a connection aided the investigation of the relationships between the linear functionals on these systems of functions and the additive set functions defined on the space. The Alexandroff spaces of this thesis are what Alexandroff himself called the completely normal spaces and what H. Gordon [1971) called the zero-set spaces

Imbeddings into topological groups preserving dimensions

AbstractWe give a negative answer to the following question of Bel'nov: Can every Tychonoff space X be imbedded as a subspace of a topological group G so that dim G ⩽ dim X? We show that if n ≠ 0, 1, 3, 7, then the n-dimensional sphere Sn cannot be imbedded into an n-dimensional topological group G (no matter which dimension function, ind, Ind or dim, is considered). However, in case dim X = 0 the answer to Bel'nov's question is “yes”. We prove that, for every Tychonoff space X, dim X =0 implies (in fact, equivalent to) dim F∗(X) = 0 and dim A∗(X) = 0, where F∗(X) (A∗(X)) is the free precompact (Abelian) group of X. As a corollary we obtain that every precompact group G is a quotient group of a precompact group H such that dim H = 0 and w(H) = w(G). A complete metric space X1 and a pseudocompact Tychonoff space X2 are constructed such that ind Xi = 0, while ind F∗(Xi) ≠ 0 and ind A∗(Xi) ≠ 0 (i = 1, 2). The equivalence of ind G = 0 and dim G = 0 for a precompact group G is established. We prove that dim H ⩽ dim G whenever H is a precompact subgroup of a topological group G. We also show that for every Tychonoff topology T on a set X with ind(X, T) = 0 one can find a precompact Hausdorff group topology T̃ on the free (Abelian) group G(X) of X such that w(G(X), T̃) = w(X, T), T̃ |x = T and dim(G(X), T̃) = 0

Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<\omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.Comment: Revised versio