关于利用vacuous truth证明空集为open ball及度量空间中空集为open set的证明疑问

Hey Linda, great question—vacuous truths can feel like a total mind-bender when you first encounter them, so it’s totally reasonable you’re confused about why your proof didn’t land with your lecturer!

First off, let’s get this straight: your proof is logically perfect. Here’s why:

• The definition of an open set in a metric space is: A set $U$ is open if for every $a \in U$, there exists an $r > 0$ such that the open ball B(a;r) is a subset of $U$.
• For the empty set ∅, there are no elements $a$ to check. In formal logic, a universal statement ("for all $x$, $P(x)$") is true when there are no $x$ to satisfy the premise—this is exactly what vacuous truth means. Since there’s no $a \in ∅$ that can violate the condition, the statement holds entirely.

Now, let’s address the two points you brought up:

Why do people call vacuous truth a "pitfall"? #

It’s mostly about intuition. We’re used to verifying statements with actual examples, so a proof that relies on "there’s nothing to disprove it" feels like a trick at first. But make no mistake—vacuous truth is a fundamental rule of first-order logic, totally rigorous and accepted in all areas of math. The "pitfall" label just refers to how easy it is to misuse or misinterpret it if you’re not comfortable with formal logic yet.

Why did your lecturer use contraposition instead? #

Your instructor probably chose contraposition (or more accurately, proof by contradiction) to avoid the confusion that vacuous truth often causes for beginners. Let’s walk through that approach to see how it connects to your proof:

Suppose ∅ is not open. By the negation of the open set definition, this means there exists some $a \in ∅$ such that for every $r > 0$, B(a;r) is not a subset of ∅.
But wait—there are no elements in ∅ at all! So this "existence" claim is impossible. Our initial assumption must be false, so ∅ must be open.

This is actually just the contrapositive/contradiction version of your vacuous truth argument. It’s the same logical conclusion, just framed in a way that feels more "concrete" because it avoids directly invoking a "vacuous" statement.

To wrap up: You didn’t misunderstand anything—your proof is valid. The pushback you got is likely just a teaching choice, not a sign your logic was wrong. If you’re still curious, feel free to ask your lecturer about the connection between the two methods; they’re two sides of the same logical coin!

备注：内容来源于stack exchange，提问作者Linda