# How to define an interior point in terms of $\epsilon$-balls?

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.

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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 $\delta=\varepsilon-d(x,y)$
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.