If a set is open, does it mean that every point is an interior point? [duplicate]

In Walter Rudin's Principles of Mathematical Analysis he defines open set as: "E is open if every point of E is an interior point of E." So this can be translated in logic as "If every point of E is an interior point of E, then E is open."

But does this mean that "If E is open, then every point of E is an interior point of E?" How is one sure when encountering these definitions that the converse also applies?

Definitions should always be treated as "if and only if". So when the text says something like "$E$ is open if every point of $E$ is an interior point of $E$" (I'm guessing it was italicized as such so as to indicate that the sentence is presenting a definition), read:

$E$ is open $\iff E$ every point of $E$ is an interior point of $E$.

Moreover, whenever you have an "if and only if" statement about an object, this statement can be used as a definition for that object. For instance, here are two possible (equivalent) definitions an author could choose for "infinite set" (there are surely many others):

$\bullet \quad$ We call a set $X$ infinite whenever there is an injection $\mathbb{N} \hookrightarrow X$.

$\bullet \quad$ We call a set $X$ infinite whenever there is a nonempty, proper subset $A \subsetneq X$ such that there is a bijection between $X$ and $X \setminus A$.