query: $\mathcal{I} = \{X \in M_{n \times n} \mid X \text{ is invertible }\}$ is not-connected.
positive_passages.docid: 114987
positive_passages.text: How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?
negative_passages.docid: 3896
negative_passages.text: A closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$
negative_passages.docid: 133266
negative_passages.text: If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis
negative_passages.docid: 4517
negative_passages.text: Showing a $2\times2$ matrix is invertible
negative_passages.docid: 31710
negative_passages.text: Show that $\mathcal{J}=\{[a, a+1) \mid a \in \mathbb{R}\}$ generates the Borel set of $\mathbb{R}$
negative_passages.docid: 148946
negative_passages.text: If $\mathbf{A}$ is a $2\times 2$ matrix that satisfies $\mathbf{A}^2 - 4\mathbf{A} - 7\mathbf{I} = \mathbf{0}$, then $\mathbf{A}$ is invertible
negative_passages.docid: 132386
negative_passages.text: Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic
negative_passages.docid: 92703
negative_passages.text: Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
negative_passages.docid: 106091
negative_passages.text: Showing $\mathcal{P}(X)$ is isomorphic to $2^X$
negative_passages.docid: 129288
negative_passages.text: proof that topologist sine curve is not locally connected
negative_passages.docid: 68724
negative_passages.text: $||T-I|| < 1$ implies that $T$ is invertible.
negative_passages.docid: 26749
negative_passages.text: $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
negative_passages.docid: 149783
negative_passages.text: $\mathbb Z\times\mathbb Z$ is principal but is not a PID
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 3296
negative_passages.text: Proof That $\mathbb{R} \setminus \mathbb{Q}$ Is Not an $F_{\sigma}$ Set
negative_passages.docid: 10042
negative_passages.text: How to Show that Linear transformation is invertible?
negative_passages.docid: 11319
negative_passages.text: Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?