My question is maybe going to be a bit vague. My apologies if so.
The setting:
Let $\overline{\rho}$ be a residual representation and $R$ be a deformation ring of $\overline{\rho}$.
Let $\mathbb{T}$ be a Hecke algebra acting faithfully on an automorphic module (let's say a subspace of the étale cohomology of a Shimura variety, for example), such that there exist a maximal ideal $\frak{m}$ and an isomorphism $R\simeq \mathbb{T}_{\frak{m}}$.
Let $L \subset \mathbf{Spec}(\mathbb{T}_{\frak{m}})$ be a locus defined by $L:=\lbrace x \in \mathbf{Spec}(\mathbb{T}_{\frak{m}}), \hspace{2mm} x \text{ satisfies the property } (\mathbf{P}) \rbrace$ where $(\mathbf{P})$ is an arithmetic property of the Galois representations $\rho_{x}$ associated to the points $x$ of $\mathbf{Spec}(\mathbb{T}_{\frak{m}})$.
We want to show topological property about the locus $L$ like "It is a Zariski open subset" or "It is contained in a codimension $n$ closed subset" or even topological property in the rigid analytic setting etc..
My question: The point of this post is that I'm looking for techniques that have been used to answer the kind of question I've raised just above. I will be really grateful to anyone who gives me references where someone manages to prove some topological property of this kind of locus $L$.
