What kind of trivial statement still needs to be proven?

Question

There are many statements that seem to be needless of a proof since they are ‘evident’ mainly because of our intuition. But some of them have proofs. For example, in C. Adams’ Introduction to topology, the Jordan Curve Theorem has a 3-page proof but the same book says, “clearly $\mathbb{R}^n$ is path connected, as is every open ball and every closed ball in $\mathbb{R}^n$" (i.e., no need to prove).

Please note that the mentioned two statements above are just examples to explain what I mean by the main question: How trivial is something trivial?

Answer 1

Not directly an answer, but too long to be a comment:

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Your example actually isn't very good. The book says:

Clearly, every open ball in $\mathbb R^n$ is path connected.

This is truly a trivial statement, since every open ball in $\mathbb R^n$ is convex, and thus path connected.

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The Jordan curve theorem, on the other hand, claims:

If $Y$ is an image of an injective continuous mapping from $S^{n}$ to $\mathbb R^{n+1}$. Then, $\mathbb R^{n+1}\setminus Y$ consists of two path connected components, of which one is bounded and the other is not. $Y$ is the edge of both components.

It's quite clear that this theorem is not equivalent to just saying "every open ball in $\mathbb R^n$ is path connected". In fact, this is a much stronger claim!

That said, of course there are simple statements that have never been proven. For example, the famous Goldbach conjecture is a simple example of a very simple statement that has not yet been proven:

Every positive even number is the sum of two prime numbers.