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I don't know much filter convergence, so this is addressed to those who do.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote a function. In elementary real analysis, we often write: $$\lim_{x \rightarrow k^+}f(x), \quad \lim_{x \rightarrow k^-}f(x), \quad \lim_{x \rightarrow k}f(x)$$

I was wondering if filters are the right way of thinking about this. If I understand correctly:

  1. Given a real number $k \in \mathbb{R}$, we get three corresponding filters, namely:
    • $\mathcal{A}(k)$, the filter generated by the intervals $\{[k,r) : k<r\}.$
    • $\mathcal{B}(k)$, the filter generated by the intervals $\{(r,k] : r<k\}.$
    • $\mathcal{N}(k)$, the neighbourhood filter of $k$.
  2. Whenever $\mathcal{F}$ is a filter on a topological space $X$ and $g : X \rightarrow Y$ is a function, we can define that $$\lim_{x \rightarrow \mathcal{F}}g(x)$$ means $\lim_Y(g_* \mathcal{F}),$ where $g_* \mathcal{F}$ is the pushforward filter.

  3. We define: $$\lim_{x \rightarrow k^+}f(x) = \lim_{x \rightarrow \mathcal{A}(k)}f(x)$$

    etc.

Question 0. Is this how it all works? If not, what have I misunderstood?

Question 1. Does anyone know of a good reference for learning the usage of filters in topology which emphasizes these kinds of concrete applications to elementary real analysis and/or functional analysis?

goblin GONE
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  • Real Analysis and Probability, by R. M. Dudley, use filter for Ascoli theorem for example. I think Bourbaki use filter a lot in their exercises. –  May 05 '15 at 15:03
  • @N.H., thanks for the suggestions. – goblin GONE May 05 '15 at 15:05
  • Depending on your definition of $\lim\limits_{x\to k^+} f(x)$ etc., you may be looking at the filters generated by the $(k,r)$, the $(r,k)$, and the punctured neighbourhoods of $k$. And the notation I'm familiar with is $\lim\limits_{\mathcal{F}} g$, or $\lim g_\ast(\mathcal{F})$. – Daniel Fischer May 05 '15 at 15:05
  • @DanielFischer, right, so we need to omit the point $k$ itself from the elements of our filters. I think this is because, if a filter $\mathcal{F}$ converges to a unique point $x$ that is furthermore an element of every member of $\mathcal{F}$, then existence of $\mathrm{lim}, g_*(\mathcal{F})$ implies the continuity of $g$ at $x$, which is obviously undesirable. Is that right? – goblin GONE May 05 '15 at 16:22
  • This seems to be related: Where has this common generalization of nets and filters been written down? Clearly, you are familiar with that question, since you commented there - but I thought that putting a link here for benefit of other users reading this question might be useful thing to do. – Martin Sleziak Dec 08 '16 at 14:40

1 Answers1

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This is really more of a comment, but I'm putting it as an answer because this list of references might be of interest to future visitors to this web page. A few real analysis textbooks discuss nets and/or filters, and I've included four such books below. The three articles I selected are notable for being slanted towards real analysis applications and for their expository merits. For example, McShane [3] is a Chauvenet Prize winning paper.

[1] Alan Frank Beardon, Limits. A New Approach to Real Analysis (1997)

[2] Theophil Henry Hildebrandt, Introduction to the Theory of Integration (1963)

[3] Edward James McShane, Partial orderings and Moore-Smith limits, American Mathematical Monthly 59 #1 (January 1952), 1-11.

[4] Edward James McShane, A theory of limits, pp. 7-29 in Robert Creighton Buck (editor) Studies in Modern Analysis, Studies in Mathematics #1, Mathematical Association of America, 1962.

[5] Edward James McShane and Truman Arthur Botts, Real Analysis (1959/2005)

[6] Herman Meyer, Introduction to Modern Calculus (1969)

[7] Herman Lyle Smith, A general theory of limits, National Mathematics Magazine [= Mathematics Magazine] 12 #8 (May 1938), 371-379.

(ADDED 1 HOUR LATER) It occurs to me that since all but Meyer [6] are reasonably well known, it might be of interest to say more about Meyer's book. The following excerpt is the 2nd paragraph of the Preface of Meyer's book. I don't have a copy of Meyer's book, and I only by chance happened to come across it in a library a few years ago (not the university library near me, but rather a library I browsed through while at a math conference a few years ago), but I did make a photocopy of some parts of the book, including the Preface.

The main portion of the book (Chapters 1 to 8) develops the concepts of the calculus, both intuitively and formally. A single theory of limits, Moore-Smith theory, unifies the entire development. Each concept involves a function having a numerical domain; a second function having a directed set as its domain is defined; and the concept is introduced by way of the limit of this second function. Presented in this form, the limit theory is adequate to cover the concepts of continuity, the derivative, the definite integral, sequences, series, and improper integrals. Proofs, throughout these chapters, are presented formally with occasional lemmas left for the student. The language and notation of sets are used freely, as are both universal and existential quantifiers. The purpose in thus developing the concepts of the calculus is to prepare the student with some gradualism for the more abstract and proof-oriented course work in linear algebra and multidimensional calculus.

Dave L. Renfro
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