# Please, as possible, explain in layman's terms: What is a discontinuous space?

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:

• It's not a discrete space.
– user122283
May 11, 2014 at 4:07
• 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.
– user14972
May 11, 2014 at 4:08
• 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. May 11, 2014 at 4:23
• 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) May 11, 2014 at 4:29
• @Jack M: Sure, but it's too long so I will edit my question. May 11, 2014 at 4:31

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.