I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to $$(x_n, f(x_n), f'_n(x_n), f''(x_n)) \to (x, f(x), f'(x), f''(x))$$ in Euclidean norm of $\mathbb R^4$...

Consider the collection of $n$ by $n$ matrices $$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$ where $c_i\in \{0,1\}$, $P_i$ and $Q_i$ are disjoint index sets, and $A(P_i;Q_i)$ is the submatrix formed by taking just the rows $P_i$ and columns $Q_i$ ...

Assume that $X$ is a topological space, and $\sim$ is an equivalence relation on $X$. Furthermore we assume that the number of elements in each equivalence class is bounded by a positive constant. Does the quotient topology on $X/\sim$ and the topology induced by the Hausdorff-metric on $X/\sim$...