Notation of a subset Is it meaningful to write $U\subset(X,\tau_X)$ where $(X,\tau_X)$ denotes a topological space? Or is it better to write $U\subset X$? Or in fact $(U,\tau_U)\subset (X,\tau_X)$?
Thanks!
 A: Even though my answer doesn't really answer the OP's question, I have to say it:  
You should of course write $U\subseteq X$.  I find $U\subset X$ just too ambiguous.  Some people use it like $U\subseteq X$, some people as $U\subsetneq X$.
$(U,\tau_U)\subseteq(X,\tau_X)$ doesn't make any sense from a set theoretic point of view, even though I agree that it is pretty clear what this means.
Similarly for $U\subseteq(X,\tau_X)$.
I think $U\subseteq X$ is appropriate, and it is usually understood that $U$ carries the subspace topology.
A: Ithink the notation $U \subset (X,\tau_X)$ should be used when it needs to be clear that $X$ is a topological space with topology $\tau_X$, otherwise - if $X$ is already defined to be a topological space - I generally use $U \subset X$. The notation $(U, \tau_U) \subset (X,\tau_X)$ defines $U$ to be a topological subspace of the topological space $X$, which is definitely different from saying that $U$ is just a subset of $X$. 
As lhf points out, if you already know that $U$ is open you can write $U \in \tau_X$.
