# Tagged Questions

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

16 views

### Using symmetry to solve Non-Linear Ordinary Differential Equations [on hold]

I know that general rules or general guidance for solving nonlinear differential equations do not exist, but im curious about the various ingenious ways that are being used to solve some of them. I ...
24 views

55 views

### In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
50 views

### Not Abelian group G with Z(G) that contains only two elements? [closed]

Is there an example of which is not Abelian group G, and Z(G) contains only two elements?
52 views

### Intermediate value property with no continuity

Definition: A real function f has the intermediate value property on an interval I containing [a,b] if f(a) < v < f(b) or f(b) < v < f(a); that is, if v is between f(a) and f(b), there is ...
42 views

### Formal definition of “proexample”. [closed]

Where in the literature do we find the preferred formal definition of “proexample” as in: the number zero is a proexample for the existential sentence "some integer is neither positive nor negative"? ...
47 views

### What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?
37 views

912 views

### Formal definition of “counterexample”.

What is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexample”...
31 views

### What is an example of lower semicontinuous functions not satisfying this?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$. Let $u:X\rightarrow [0,\infty]$ be a lower semicontinuous function such that $\int_X u d\mu <\infty$. Then, does ...
44 views

### Examples of $R$-modules $X$ such that $(X \setminus TX) \cup \{0\}$ isn't a submodule.

Work over an ambient commutative ring with unity. Given a module $X$, write $TX$ for its submodule of torsion elements. Suppose we want to find the "submodule" of torsion-free elements of $X$. So ...
32 views

73 views

### Counterexamples in Analysis

I want to (dis)prove the following statement: A sequence of functions which converges almost uniformly implies uniform convergence for that sequence of functions. I'm sure I've read up on a ...
80 views

### Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
23 views

25 views

### Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ :  dist\big(\gamma(s), \gamma(t)\...
35 views

### Are there groups $G$ isomorphic to $\mathrm{Aut}\left( \mathrm{Aut} (G)\right)$, such that $G\not\cong \mathrm{Aut}(G)$?

Trivial examples for the first condition are easy to find: $G={1},C_2$. Are there any (finite\non-finite) groups that satisfy the conditions in the title?
54 views

### Define any non-commutative operation for the group $\left({\mathbb{R}, \circ}\right)$ [closed]

Let $\mathbb{R}$ denote the set of real numbers. Given that $\left({\mathbb{R}, \circ}\right)$ is a group, provide any definition for $\circ$, so that $\circ$ is not commutative.
1k views

### A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
70 views

### Disprove the statement: If $g\circ f=I_X$then $f\circ g=I_Y$. [closed]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: Let $f : X \rightarrow Y$ and $g : Y \rightarrow X$ be functions. If $g\circ f=I_X$...
109 views

### are continuous functions that map measure zero sets to measure zero sets absolutely continuous?

Let $I$ be a closed interval and $f:I\rightarrow \mathbb{R}$ be a continuous function which maps measure zero sets to measure zero sets. If $f$ is monotonically increasing, $f$ must be absolutely ...
29 views

### If a sequence converges in measure, are convergent subsequences of it all converge to the same limit?

Let $f_n:X\rightarrow \mathbb{C}$ be a sequence of measurable functions such that $f_n\rightarrow f$ in measure. Let $f_{n_k}$ be a subsequence of $f_n$ such that $f_{n_k}\rightarrow g$ pointwise a.e....
### Disprove the statement $f(A \cap B) = f(A) \cap f(B)$ [duplicate]
If someone could walk me through this I would greatly appreciate it. Disprove the following statement: If $f : X \rightarrow Y$ is a function and $A$, $B$ are subsets of $X$ then \$f(A \cap B) = f(A)...