Intereting Posts

Lines cutting regions
Regularity of a quotient ring of the polynomial ring in three indeterminates
Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K
Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?
For two problems A and B, if A is in P, then A is reducible to B?
Prove that 3 is a primitive root of $7^k$ for all $k \ge 1$
Division algorithm of multivariate polynomial
What map of ring spectra corresponds to a product in cohomology, especially the $\cup$-product.
How to find all subgroups of a group in GAP
$a,b,c,d\ne 0$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $
Show that if a function is not negative and its integral is $0$ than the function is $0$
Integration of $\dfrac{x}{\sinh x}~dx$ from $-\infty$ to $\infty$
What sets are Lebesgue measurable?
Subgroups of finite index in divisible group
Computing Ancestors of # for Stern-Brocot Tree

Let $M$ be the space of all $m\times n$ matrices. And $C=\{X\in M|\operatorname{rank}(X)\leq k\}$ where $k\leq \min\{m,n\}$. Check whether the set $C$ is:

- Closed
- Connected
- Compact
- Open

What are some other good properties of the set $C$,for example is it a manifold?

Clearly the set $C$ is closed if someone is interested a good proof can be found here, hence $C$ is not open. Also as $C$ is unbounded therefore not compact. How to check whether the set $C$ is connected or not?

- Proof of the Borsuk-Ulam Theorem
- Product topology and standard euclidean topology over $\mathbb{R}^n$ are equivalent
- Which sequences converge in a cofinite topology and what is their limit?
- Is there a continuous bijection between an interval and a square: $ \mapsto \times $?
- Can a space $X$ be homeomorphic to its twofold product with itself, $X \times X$?
- Why does Van Kampen Theorem fail for the Hawaiian earring space?

- How do I prove that in every commuting family there is a common eigenvector?
- Shrinking Group Actions
- Proof check/ suggestion: The suspension of $S^n$
- How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?
- Proving determinant product rule combinatorially
- Good metric on $C^k(0,1)$ and $C^\infty(0,1)$
- Proving a certain inequality
- Is this a perfect set?
- Topology textbook with a solution manual
- Is integrability of a function a local property?

If $M \in C$, then $\lambda M \in C$ for all $\lambda$, hence $M$ and $0$ are path connected. It follows that $C$ is connected.

Pick a non zero $M \in C$, then $\lambda M\in C$ for all $\lambda$, hence $C$ is not bounded, hence not compact.

If $k=\min\{m,n\}$, then see easily that $C$ is open and closed, so suppose $k < \min\{m,n\}$.

Let $\mu_1,…,\mu_p$ be the linear maps that correspond to all the $k+1$ minors of a $m \times n$ matrix, and let

$\phi(M) = \min(\det \mu_1(M),…,\det \mu_p(M))$. Then we see that

$M \in C$ **iff** $\phi(M) = 0$, and since $\phi$ is continuous, we see that

$M = \phi^{-1}\{0\}$ is closed.

Now suppose $M \notin C$, that is, $M$ has rank $k+1$ or greater. Then

${1 \over k} M \notin C$, and ${1 \over k} M \to 0 \in C$, hence the complement $C^c$ is not closed. It follows that $C$ is not open.

- Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$.
- Calculating a value inside one range to a value of another range
- Arrange $n$ people so that some people are never together.
- How to calculate integral of $\int_0^\sqrt4\!\sqrt\frac{x}{4-x^{3/2}}\,\mathrm{d}x$
- A conjecture including binomial coefficients
- Intersections of a ray from star domain to its boundary
- How to prove that geometric distributions converge to an exponential distribution?
- For given prime number $p \neq 2$, construct a non-Abelian group with exponent $p$
- If $\Gamma \cup \{ \neg \varphi \}$ is inconsistent, then $\Gamma \vdash \varphi$
- Beginner's book for Riemannian geometry
- Zeros of Fourier transform of a function in $C$
- Prove that an infinite ring with finite quotient rings is an integral domain
- How to get the aspect ratio of an image?
- Showing that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges (Baby Rudin Exercise 6.9)
- Does $i^4$ equal $1?$