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This may not completely answer your question, but it sounds like what you're talking about is related to the concept of [topological embedding](https://en.wikipedia.org/wiki/Embedding). A function $f: X\to Y$ that is a homeomorphism from $X$ to $f(X)$ is called an embedding. For a given $X$ and $Y$, the existence of an embedding of $X$ in $Y$ is a topological invariant. This is a relationship between sets though. Your relationship could be stated in terms of embedding as follows:

 > Given two points $p\in X$ and $q\in Y$, we have $p\precsim q$ if there is an embedding $f$ from a neighborhood of $p$, into $Y$, with $f(p)=q$.