Articles of nets

Properties of generalized limits aka nets

I want to find some article or a book which contains all general properties of nets. Of course some of them similar to properties of sequences with almost the same proofs, but I don’t fill the edge, where nets and sequences starts to behave differently. Also it would be nice to find complete survey on […]

Introductory text on nets

I have learned point-set topology using filters. Now I do functional analysis where we are using nets to do topological stuff. Therefore I search an introductory text on nets that is suitable for this purpose, i.e. to lay the foundations for usage in FA and maybe that the text assumes some knowledge in point-set topology, […]

Nets and Convergence: Why directed indices?

Please do read carefully (I know Nets-Topology-Filters and their interrelations!!!) 1.) Why do we require nets to be indexed by directed sets (apart from it simply works compared to filters and topology). Is there a reason w.r.t. to the notion of convergence? So far there are those hints: Notion of a good Direction Uniqueness of […]

Why are nets not used more in the teaching of point-set topology?

I just finished working through a proof of Tychonoff’s Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components do). While I might be missing steps (I based the proofs off some optional exercises in a textbook, but the […]

What's the “limit” in the definition of Riemann integrals?

Consider one of the standard methods used for defining the Riemann integrals: Suppose $\sigma$ denotes any subdivision $a=x_0<x_1<x_2\cdots<x_{n-1}<x_n=b$, and let $x_{i-1}\leq \xi_i\leq x_i$. Then if $$|\sigma|:=\max\{x_i-x_{i-1}|i=1,\cdots,n\},$$ which we shall call the norm of the subdivision, we define: $$\int_a^bf(x)dx:=\lim_{|\sigma|\to 0}\sum_{i=1}^nf(\xi_i)(x_i-x_{i-1}).$$ When one talks about the limit of a function $\lim_{x\to x_0}f(x)$, one has exactly one value […]

Weak net convergence in $\ell_p$, where $1 < p < \infty$.

EDIT: The question is edited after an error pointed out by gerw. There are the following two results regarding weak convergence in $\ell^p$ spaces: Let $((\beta_n^{(\alpha)}))_{\alpha \in I} \subseteq \ell_p (\mathbb{N})$ be a net and $(\beta_n) \in \ell_p (\mathbb{N})$, where $1 < p < \infty$. Then (i). $\beta_n^{(\alpha)} \to \beta_n$ for each $n \in \mathbb{N}$ […]

Subnets and finer filters

Suppose $G$ is a finer filter than $F$ in a topological space $X$. Is the net base in $G$ a subnet of the net base in $F$? I am using the definitions of General Topology of Willard: Definition 12.15. If $(x_\lambda)$ is a net in $X$, the filter generated by the filter base $\mathscr C$ […]

Where has this common generalization of nets and filters been written down?

It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters. They are of course essentially equivalent, but each has its own minor advantages for pedagogy and intuition. Seemingly less well-known is the following common generalization of both nets and filters: Let […]

Example of converging subnet, when there is no converging subsequence

I’m trying to wrap my head around the concept of nets/subnets, especially in the following example. Let $X$ be the Banach space $\ell_{\infty}$ and $X^*$ its dual. We know by Banach-Alaoglu that the unit ball $B$ of $X^*$ is compact, but not metrizable (as $X$ is not separable). It is compact but not sequentially compact, […]

Every subnet of $(x_d)_{d\in D}$ has a subnet which converges to $a$. Does $(x_d)_{d\in D}$ converge to $a$?

Let $(X,\mathcal T)$ be a topological space and $(x_d)_{d\in (D,\le)}$ be a net in it and let $a\in X$. Every subnet of $(x_d)_{d\in D}$ has a subnet which converges to $a$. Does $(x_d)_{d\in D}$ converge to $a$? In fact, $a$ is a cluster point of every subnet as defined in this wikipedia page.