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What is a "discontinuous space"? Is it synonymous of "discrete space"?

I searched in Google but did not find an accessible explanation. I have an idea of it as a space where all lengths are multiples of some "elementary" value, but I'm not sure if it's this or how geometry works in such a setting (what become of the theorems I know for example).

I'm asking because I read recently in a discussion forum (In Portuguese) that the Pythagorean theorem is false in any type of discontinuous space. I did not understood very well what it meant (so I did two searches after reading that), but I got very curious about "discontinuous spaces" and how is geometry in them.

I would like answers that don't involve too much advanced topics, but they are welcome too (although I will not be able to understand them :), haha)

P.S. I've already read this article, but did not understand its definition: "a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.".

As required, I'm citing the original statement that I mentioned:

Se aceitar o método da Lógica Formal para investigação da verdade, então num espaço ortonormal contínuo Minkowskiano, um triângulo retângulo desenhado sobre uma superfície plana deste espaço sempre terá uma uma hipotenusa cujo comprimento elevado ao quadrado será igual à soma do comprimento de cada cateto elevado ao quadrado. É importante adotar a premissa de que o espaço seja Minkowskiano, já que este teorema seria falso num espaço Lobachevskiano, num espaço Riemanniano, num espaço fractal, em qualquer tipo de espaço descontínuo, em espaços não ortonormais (como os que exigem métrica de Kerr-Newman). Enfim, tomando os devidos cuidados na seleção dos axiomas e na formulação da declaração, ela pode expressar uma verdade "absoluta". "

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    $\begingroup$ It's not a discrete space. $\endgroup$
    – user122283
    May 11, 2014 at 4:07
  • $\begingroup$ I've never heard of "discontinuous space" before. I'm familiar with disconnected, though, and my first guess at hearing the phrase is that disconnected is what was meant. $\endgroup$
    – user14972
    May 11, 2014 at 4:08
  • $\begingroup$ I saw now that one can read there only with register... But since it's Google owned, the cache version shows the page msgs: webcache.googleusercontent.com/… The word used was the Portuguese "descontínuo" which according to Google Translate can be in English: "discontinuous", "discrete" or "discontiguous". But I'm starting to believe that it was a typo or something, because of what you said too. $\endgroup$
    – João Rimu
    May 11, 2014 at 4:23
  • $\begingroup$ I don't see any mention of the Pythagorean theorem on that page. Since it seems whoever made the statement you're referring to wasn't using standard terminology, I think we really need more context here. Could you include, say, the paragraph surrounding the statement that you're citing? (even in the original portugese) $\endgroup$
    – Jack M
    May 11, 2014 at 4:29
  • $\begingroup$ @Jack M: Sure, but it's too long so I will edit my question. $\endgroup$
    – João Rimu
    May 11, 2014 at 4:31

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The Pythagorean Theorem is false in a "discontinuous" space because in a "discontinuous" space you are not guaranteed that the equality can hold.

The Pythagorean Theorem says that $a^2+b^2=c^2$. Suppose you are working in the space of integers. Take the example of a triangle $\sqrt2+\sqrt2=2^2$. This is not allowed in the space of integers because $\sqrt2$ is not an integer. So if you want a triangle with a hypotenuse of length 4, you can't have it in the space of integers.

EDIT:

I ran the quote through Google Translate and it occurred to me that you first need to understand the difference between different kinds of spaces. Spaces that are not complete are problematic (limits of functions don't necessarily exist, solutions to equations don't necessarily exist, etc). I can't answer the physics portion of your question because it is too broad, and I'm not sure what you're looking for. If you don't understand what a complete space is, I suggest you find a Portuguese book on advanced mathematical analysis and then return to your original problem. (I don't know if the Walter Rudin book is translated in Portuguese, but undoubtedly something similar exists.)

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  • $\begingroup$ Given the keywords in the Portugese quote (Minkowski space, Riemannian space) this is clearly not even close to the intended meaning. $\endgroup$
    – Jack M
    May 11, 2014 at 5:04
  • $\begingroup$ My explanation was in reference to this paper he cited. krmcdani.mysite.syr.edu/DD.pdf As stated, the question is vague: "how is geometry in discontinuous spaces". I specifically dealt with the Pythagorean Theorem because that's the only specific geometry thing he asked about. I don't speak Portuguese so I don't know what the quotation he mentioned is saying. $\endgroup$
    – rocinante
    May 11, 2014 at 5:11

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