Which Topological Spaces are Powers?

Given a topological space X and closed subspace Y⊂X, it may be the case that X is a power of Y. That means X=∏i<κYi for some cardinal κ where each Yi≅Y. Are there any internal conditions on how Y sits inside X that guarantees it is a factor? Has any research been carried out as … Read more

Examples of a topological semidirect product

Let G be a compact topological group, and Aut(G) the group of autohomeomorphisms of G. I have proved some (topological) results about the holomorph G⋋, and now looking for nice examples. I can, of course, take any compact group G, but I want to know if there are “natural” examples to be found. That is, … Read more

algebraic structure of Integral Steenrod squares

It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations $$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$ In the case where $a$ is odd, one can define an integral refinement $$Sq_{\mathbb{Z}}^a=\beta(Sq^{a-1})\rho_2$$ where $\beta$ is the Bockstein corresponding to the sequence $$\mathbb{Z} \overset{\times 2}{\to} \mathbb{Z} \overset{\rho_2}{\to} \mathbb{Z}/2.$$ This does indeed refine the usual Steenrod squares since … Read more

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the following generalization: For three given points $a,b,c \in \mathbb{R}^{2}$ define $$A_{\lambda}=\{z\in \mathbb{R}^{2} \;\text{with}\;\; |z-a|+|z-b|+|z-c|=\lambda\}$$ How is the geometric description of … Read more

Is the following product-like space a Polish space?

Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-Prokhorov metric. For $\mu \in \mathcal{M}_1(\mathbb R)$, let $L^1(\mu)$ denote the Banach space of $\mu$-integrable functions (mod $\mu$-null). Again, for each $\mu$, $L^1(\mu)$ is a … Read more

Intuition for universal quotient maps

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), the two characterizations of universal quotient maps below are mentioned. Proposition. For a continuous map p, TFAE. p is a universal quotient map (i.e., a regular epi … Read more

Topology on the space of Borel measures

Let B be the set of all measures ϕ of Rn such that every open set is ϕ-measurable (sometimes these measures are called Borel measures). Note the measures in B are not required to be locally finite. Let 1≤m≤n be an integer. The upper density of a measure ϕ at a∈Rn is defined as Θ∗m(ϕ,a)=lim sup … Read more

Homeomorphism between evenly spaced integer topology and the rationals

The evenly spaced integer topology is countable, metrizable, and has no isolated points, and hence is homeomorphic to the rationals with the order topology. But what is an explicit construction for this homeomorphism? This question was asked on MSE but got no answer: https://math.stackexchange.com/q/1849271/52694 Answer AttributionSource : Link , Question Author : Mike Battaglia , … Read more

Counting loops in degree: 1 or 2?

Here’s what seems to be an annoying technicality when dealing with loops in graphs. In the literature on expander graphs (and surely not only), it seems to be the convention that a loop at vertex v in an undirected graph contributes 1 to the degree, and 1 to the corresponding diagonal entry in the adjacency … Read more

Basic calculus on topological fields

Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $\mathbb Q_p$). 1) Let $f: K^n \to K$ be a function such that its differential $Df$ is constant $0$. Is $f$ constant on … Read more