(I went through some related StackExchange questions but none addresses the one I have. If you do find one that addresses mine below, I'd be happy to look into that.)
In Chapter 6.2 of Casella and Berger's Statistical Inference where the topic of sufficient statistics is discussed, I couldn't understand a statement that has been mentioned a few times:
... Note that events {$\mathbf{X} = \mathbf{x}$} ... subsets of {$\mathit{T}(\mathbf{X}) = \mathit{T}(\mathbf{x})$}.
I understood (I think) that, with a sample space $x \in \mathcal{X}$, a statistic (transformation function) $\mathit{T}$, and the image $T(x) \in \mathcal{T}$, there is a partition of elements in $\mathcal{X}$ that corresponds to an element in $\mathcal{T}$. What I don't understand is, how is {$\mathbf{X} = \mathbf{x}$} a subset of {$T(\mathbf{X}) = T(\mathbf{x})$} if they're not in the same space?