Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Which is the technically correct definition?

I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$.

II) An interior point of a set $B$ is a point that is in a set $A\subset B$ in which every point is the centre of some $\epsilon$-ball in $A$.

The two definitions (which I made up myself) don't seem equivalent, and I can't figure out which is the incorrect definition. Here, please do not define interior point in terms of neighbourhoods or open sets.

share|improve this question

2 Answers 2

up vote 2 down vote accepted

They are equivalent:

If $x\in B$ is the center of $B_\varepsilon(x)\subseteq B$, then each $y\in B_\varepsilon(x)$ is the center of a ball $B_\delta(y)\subset B_\varepsilon(x)$ if you choose $\delta=\varepsilon-d(x,y)$

share|improve this answer

The two definitions are equivalent. Suppose that $B(x,\epsilon)\subseteq B$, and let $y\in B(x,\epsilon)$; then you can use the triangle inequality to show that $B\big(y,\epsilon-d(x,y)\big)\subseteq B(x,\epsilon)$, thereby showing that (I) implies (II). The reverse implication is clear.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.