It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the second paragraph, highlighted with black background, as shown in this image.
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2$\begingroup$ Does this answer your question? What is a universal tree? $\endgroup$– Ryan BudneyCommented Jan 15, 2023 at 6:38
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$\begingroup$ @RyanBudney, the OP is asking about $\mathbb{R}$-trees, which are a different kind of trees. $\endgroup$– Kostya_ICommented Jan 15, 2023 at 12:22
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1$\begingroup$ That's a little sad. Apparently "universal trees" are not universal. $\endgroup$– Ryan BudneyCommented Jan 15, 2023 at 20:13
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2 Answers
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In the beginning of Section 6 on p. 318 of Menger's Kurventheorie, referred to in Kolmogorov's paper on the page whose image you linked in your post, we find this:
Wir bezeichen eine Baumkurve $B$ als einen Universalbaum bzw. als Universalbaum $n$-ter Ordnung, wenn $B$ zu jedem vorgelegten Baum bzw. zu jedem vorgelegten Baum $n$-ter Ordnung eine homöomorphe Menge als Teilmenge enthält.
which apparently means
We refer to a tree curve $B$ as a universal tree, or a universal tree of the $n$th order, if $B$ contains as a subset a homeomorphic image of any given tree, or any given tree of the $n$th order.
answered Jan 15, 2023 at 14:26
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$\begingroup$ I think you're right,but when i read the passage,it's too hard to find the electronic edition of second quoted passage ,where did you find it? $\endgroup$– ooo mmmCommented Jan 16, 2023 at 14:56
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$\begingroup$ @mmmooo : I quoted only one passage. The second highlighted piece of text is just my attempt at translation (aided by Google Translate) of the first highlighted piece of text (in German, which I don't know). $\endgroup$ Commented Jan 16, 2023 at 15:03
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$\begingroup$ I mean to say,where did you find the k. menger's book kurventheorie,which is the second quotation of the Kolmogorov's paper.😂 $\endgroup$– ooo mmmCommented Jan 16, 2023 at 16:27
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$\begingroup$ Could you give me a download link? $\endgroup$– ooo mmmCommented Jan 16, 2023 at 16:29
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$\begingroup$ @mmmooo : I used archive.org/details/kurventheorie0000karl/page/n9/mode/… . In your case, the archive.org link may be different. $\endgroup$ Commented Jan 16, 2023 at 16:32
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I'm here to answer my own question,In the ending of Section 1 on p. 306 of Menger's Kurventheorie ,which expound the concept in a modern way.
Literatur. Der Begriff der Baumkurve: Mazurkiewicz (Fund. Math. 2, 1921, S. 123). Erblicher Zusammenhang im Kleinen der Baumkurven: Ważewski (Ann. Soc. Polon. Math. 2, 1924, S. 83), Scherrer (Math. Ztschr. 24, 1926, S.127). Regu- larität der Baumkurven: Menger (Math. Ann. 96, 1927, S. 573). In der amerika- nischen Literatur werden Baumkurven meist als acyclic continuous curves bezeichnet, im Französischen als dendrite.
Literature. Concept of tree curve: Mazurkiewicz (Fund.Math.2, 1921, p. 123). Genetic relationship of small tree curve: Wazewski (Ann. Soc. Bronn. Mathematics. 2, 1924, p. 83), Scherrer (Math. ZTCHR. 24, 1926, p. 127. Regularity of tree curves: Menger. Ann. 96, 1927, p. 573. In American literature, tree curves are usually called acyclic continuous curves and dendrites in French.
You can use the link from losif Pinelis above. But the new problem come out,which book can replace the kurventheorie ,the German and scanned version ebook make the reading so hard.
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$\begingroup$ @Iosif Pinelis would you notice this? $\endgroup$– ooo mmmCommented Feb 6, 2023 at 16:06
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$\begingroup$ Kolmogorov's paper is, of course, very old. So, maybe, there is no such replacement, despite the inconveniences of reading Menger's book. Upvoted your answer nonetheless. $\endgroup$ Commented Feb 6, 2023 at 16:22