Well, point-set topology is hardly my métier, but it seems to me that when one speaks of a set being closed, one always has in mind its setting within a larger space. One says (or ought to say), “$K$ is closed in $X$”, because the property of being closed is not intrinsic.
Once you understand that, you see that the property of being open is not intrinsic, either. Here's an example. In the space $X=[0,1]$, that is, $X$ is the closed unit interval, the subset $U=\langle1/2,0]$, that is the half-open interval from $1/2$ to $1$, is open. But it is not open in $\mathbb R$. About the point $1\in X$, the “$X$-ball” of radius $1/4$ is wholly contained in $U$, because the $X$-ball of radius $1/4$ contains no points to the right of $1$.
There is a far deeper result that says that an open subset of $\mathbb R^n$, when mapped continuously and in one-to-one fashion into another $\mathbb R^n$ (same dimension), has image that is necessarily open. This is “Invariance of Domain”, which you can google, and it does say that in this restricted sense, openness in $\mathbb R^n$ is intrinsic. But as I say, that is a deep theorem.