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Of course $f^\beta(p)$ is the limit of $f(x)$ along the z-ultrafilter $\mathcal A^p$. For $t \in \mathbb R$, we write $$ \lim_{\mathcal A^p} f(x) = t $$ iff for every neighborhood $U$ of $t$ there is …

If $V$ or $W$ finite-dimensional, then $V \otimes W$ consists of sums $$\sum_{j=1}^n v_j\otimes w_j\tag1$$ with a fixed number $n$ of terms. But when $V,W$ are infinite-dimensional, you have sums $\su …

I think there is a problem with a mere semi-norm. The constant functions have variation distance $0$ from each other. Any variation-open set contains either all the constants, or none of them. There …

Another case. Let $B_1, B_2, \dots$ be compact sets in $\mathbb R^n$, all bounded by $R > 0$. Let $w_n$ be nonnegative numbers with sum $1$. The product topological space $X = B_1 \times B_2 \times …

Your criterion is (half of) uniform convergence. As commented, not every function can be uniformly approximated by Riemann integrable functions. I say "half" because you wrote $f(x) -R(x) < \epsilon$ …

No. Note that if field $F$ has topology induced by $[0,\infty)$-valued absolute value, then either $\{x \in F : |x|<1\} = \{0\}$ so that the topology is discrete, or else there is $x$ with $0 < |x| < …

Say we have a partially ordered set. What do you doubt? (1) The set of intervals $\left[x,\rightarrow\right[$ is a base for a topology. (2) Any intersection of open sets is open. (3) The closure o …

Bockstein's theorem Bockstein, M., Un théorème de séparabilité pour les produits topologiques, Fundam. Math. 35, 242-246 (1948). ZBL0032.19103. This is the case of a product $\prod_{t \in T} X_t$ wher …

We will use the Kuratowski–Ryll-Nardzweski selection theorem: Let $(\Omega, \mathscr{F})$ be a measurable space. Let $E$ be a Polish space. Let $\Gamma$ be a set-valued function from $\Omega$ to $E$ …

Remarks and hints, not a solution Question 1 A disjoint union $\mathcal F = \bigcup_{i \in I} \mathcal F_i$ in a metric space has the disjoint union topology if and only if all sets $\mathcal F_i$ ar …

No, not in general. My metric space is the disjoint union of uncountably many copies of $\mathbb R$. $$X = \bigsqcup_{t \in T} X_t$$ where $T$ is uncountable and $X_t = \mathbb R$ for all $t$. The …

Perhaps look for "uniform spaces" ... this is a generalization of metric spaces, with no need to use real numbers in the definition. Wikipedia has a page on uniform spaces. Many textbooks on point-s …

comment I found HERE A space is called screened if every open covering has a $\sigma$-disjoint open refinement Do you think this is equivalent to weakly Lindelof: every open covering has a $\si …

If $U= (-\infty,0) \times \mathbb R \times \mathbb R \times \dots$, then $\mathrm{cl}(U) = (-\infty,0] \times \mathbb R \times \mathbb R \times \dots$, and the complement of this is $(0,+\infty) \tim …

Searching Steen & Seebach, Counterexamples in Topology for "Hausdorff, separable, not regular" I found this example. Is that it? This is Example 79, page 97. Irregular Lattice Topology Let $ …