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 ...

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1answer
61 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
6
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1answer
350 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in ...
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1answer
51 views

Counterexample to conditional probability with dependent events

Let $X1,X2,X3$ be i.i.d. taking values in a finite set, and not constant. Is it necessarily true that $P(X3=X2|X2≠X1)≤P(X3=X2)$? Give a proof or a counterexample. Since the two events $A=\{X3=X2\}$ ...
3
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0answers
36 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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0answers
24 views

Example for Stieltjes Integral? [on hold]

I have problem make a example from this paper about Stieltjes Integral and you can check at stieltjes integral like this :
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1answer
23 views

Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
31
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9answers
6k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
6
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1answer
189 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
0
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1answer
33 views

commuting subsets in a group

For a countable infinite discrete group $G$, consider the following three properties. (P1) $G$ is abelian. (P2) For any finite subset $K$ of $G$, there exists an element $s\in G$, such that the two ...
3
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1answer
165 views

Example of $f,g: [0,1]\to[0,1]$ and riemann-integrable, but $g\circ f$ is not?

Give me an example of two Riemann-integrable functions $f,g:[0,1]\to[0,1]$ such that $g\circ f$ isn't integrable! I already know the following example: $$f(x)=\begin{cases} 0, & \text{if ...
2
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1answer
35 views

Is every open set in a base space evenly covered?

Let $C$ be a covering space of $B$. Then, does every open set in $B$ evenly covered by a covering map? This must be false but I cannot find a counterexample.. Please help
2
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0answers
64 views

Counterexample to an implication

Denote $\bar{A}$ a complement of $A$ in a set $\Omega$ and $A \Delta B = A/B \cup B/A$ the symmetric difference of $A, B$. It is claimed that for a map $\phi := \Omega \rightarrow \lbrace 0, 1 ...
2
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1answer
31 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
8
votes
4answers
286 views

Unconventional (but instructive) proofs of basic theorems of calculus

Inspired by this questions asked on MathOverflow, I would like to ask if you know some "sophisticated" proofs of the basic theorems in a calculus course (that is, the ones that you can find, for ...
1
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3answers
51 views

Find two functions $f$ and $g$ such that they are both discontinuous at $c$, however, $f+g$ and $f\cdot g$ are both continuous at $c$

Could someone please explain to me how to approach these kinds of question and also what is the answer to the following question? Give an example of a function $f$ and $g$ such that they are both ...
1
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1answer
43 views

Open (but not closed) subgroups of $GL_n$

The book I am currently reading states: "...as we will see later, non-closed subgroups [of $GL_n(\mathbb K)$] are not necessarily manifolds." Prompted me to think about open subgroups of $GL_n$: ...
1
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1answer
48 views

What is a counterexample for this one?

Let $x$ be an irrational number. Let $\{a_0\}$ be the sequence of positive integers except for $a_0$ such that $x=a_0+K(1/a_n)$. Let $a,b$ be integers such that $b>0$ and $gcd(a,b)=1$ and ...
1
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0answers
27 views

Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
0
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2answers
26 views

The product of two nonnegative, improperly integrable functions is also improperly integrable.

True or False: The product of two nonnegative, improperly integrable functions is also improperly integrable. I was given both the problem and the proof that may or may not be true. I think the ...
4
votes
2answers
41 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
1
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1answer
24 views

Show that the space of superharmonic functions is not a linear space

Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $P=(p_{i,j})_{i,j\in E}$. A real valued function $h$ on $E$ is called superharmonic if $h(x)\geq Ph(x)$ ...
1
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1answer
21 views

Lebesgue integrable discontinuity points

If a function is Lebesgue integrable, is it possible that it has as set of discontinuity points measure bigger than zero?
3
votes
3answers
90 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
1
vote
1answer
40 views

Arrange 10 points on five lines where each line(intersecting) has exactly 4 points

One possible case is that forming a star and then arranging 10 points on its vertices. Is there any other possible case for this arrangement? If not then how can we prove it mathematically? ...
4
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1answer
136 views

Generalized Riemann Integral: Bounded Nonexample?

Reference For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For an improper version of integral see: Riemann Integral: Improper Version For a comparison of integrals ...
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1answer
42 views

Compactness of second-countability of $\omega$X$\omega_1$

Please discuss the following properties of the product space consisting of $\omega$X$\omega_1$: Is it compact? Is it 2nd countable? $\omega$ is the first infinite ordinal and $\omega_1$ is the ...
5
votes
1answer
44 views

An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$

I am thinking about this problem: Let $f\in L^1 [0,1]$ to be a nonnegative function satisfied: $$\int_{E} f dm\leq \sqrt{m(E)}$$ for every measurable set $E\subset [0,1]$, Prove that $f\in ...
1
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3answers
49 views

Complex Mean Value Theorem: Counterexamples [closed]

This thread is just to collect some examples... Given an open domain $\Omega\subseteq\mathbb{C}$. Consider a holomorphic function $f:\Omega\to\mathbb{C}$. What would be a counterexample to: ...
1
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0answers
19 views

Why is it so difficult to obtain the spectral properties related to infinite matrices, especially when they are not symmetric?

I believe most of the spectral theory is revolving around the bounded self-adjoint linear operators being analogous to real symmetric infinite matrices. Whereas, there are cases when the matrices are ...
0
votes
1answer
85 views

Are the following topological spaces locally compact?

I am trying to determine whether the following spaces are locally compact: a) the slotted plane b) the radial plane For part a) I am almost certain that it is not compact, but not sure how to go ...
1
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2answers
29 views

Non-unital commutative semigroups $S$ such that for all $x \in S$, the function $S \rightarrow S$ given by $y \mapsto x+y$ is a bijection?

Does there exist a commutative semigroup $S$ with the following (additively denoted) properties? For all $x \in S$, the function $S \rightarrow S$ given by $y \mapsto x+y$ is a bijection. $S$ has no ...
8
votes
2answers
139 views

The topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$.

I'm looking for an example of a topological space $X$ together with an equivalence relation $\sim$ where the product topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$ as a final ...
2
votes
1answer
22 views

$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Let $F$ be a field and $K$ be an extension field of $F$. Proof\Counterexample: $K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$. I haven't ...
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0answers
25 views

Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
3
votes
2answers
73 views

A topological space which is Frechet but not Strictly-Frechet.

Let $X$ be a topological space and $q \in X$. $X$ is strictly Frechet at $q$, if, for all $A_n \subset X, q \in \bigcap_{n \in \omega} \overline {A_n}$ implies the existence of a sequence $q_n \in ...
1
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1answer
20 views

Example of an increasing non-nonnegative sequence violating conclusion of monotone convergence theorem in space of finite measure

With Lebesgue measure in $\mathbb{R}$, $f_n(x) \equiv -\frac{1}{n}$ is a good example which doesn't coincide with MCT. However, I couldn't find another example when the measure is finite. Could ...
1
vote
1answer
81 views

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this ...
1
vote
1answer
35 views

Metrizable compact spaces and Hausdorff spaces with a countable network

I have two questions related to metrizable spaces and countable network ; Can we find a continuous mapping from a separable metric space onto a non metrizable compact Hausdorff space. If a Hausdorff ...
0
votes
0answers
27 views

Necessary and sufficient condition

could someone help me with 'necessary and sufficient' conditions problem with full proofs or counterexamples. thank you. It's either a 'necessary', 'sufficient', 'necessary and sufficient' or ...
0
votes
1answer
42 views

Cantor Intersection Theorem Without Closedness, counterexample

The Cantor Intersection Theorem is that Let $\{S_1,S_2,S_3,...\}$ be a countable collection of nonempty sets in $\mathbb R$ such that: $S_{k+1} \subset S_k$ for $k=1,2,3...$ Each $S_k$ ...
7
votes
2answers
115 views

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$?

How to determine the existence of the function $f: \mathbb{R} \rightarrow \mathbb{R}$, which is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$? It's more than enough to give an ...
0
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1answer
61 views

Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?
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0answers
47 views

Bochner Integral vs. Lebesgue Integral

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
4
votes
2answers
55 views

A category with arbitrary products, but not all limits, or finite limits not commuting with filtered colimits?

I'm interested in finding an example of a locally small category $\mathcal{C}$ having small filtered colimits and arbitrary small products but lacking, either all small limits, or either the ...
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votes
2answers
53 views

Why does differentiability implies continuity, but continuity does not implies differentiability?

Why does differentiability implies continuity, but continuity does not implies differentiability? I am more interested in the part about a continuous function not being differentiable. Well, all ...
1
vote
0answers
21 views

Flow-Invariance w.r.t. $A$ doesn't invariance w.r.t. $\exp (tA)$

This answered question of mine explains that by saying consider $$\dot x =Ax,$$ then the stable and unstable subspaces are invariant with respect $A$ and therefore also with respect to $\exp (tA)$ one ...
0
votes
0answers
10 views

Iterated limits difficult example

Is there a fucntion $f:[-1,1]\subseteq \mathbb R^2 \to \mathbb R$ so that $$lim_{x\to 0}f(x,y)$$ exist $\forall y_0\in [-1,1]$ fixed and and $$lim_{y\to 0}f(x,y)$$ exist $\forall x_0\in [-1,1]$ ...
3
votes
2answers
24 views

Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $lim_{x\to x_0}f(x,y)$ does not exist

Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $$lim_{y\to y_0}f(x,y)$$ and $$lim_{x\to x_0}(lim_{y\rightarrow y_0}f(x,y))$$ exists but $lim_{x\to x_0}f(x,y)$ does not ...
14
votes
1answer
262 views

Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
1
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2answers
35 views

Nonmeasurable Functions

Reference This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral Problem What are examples of real-valued functions: Bounded & Non-Step & Non-Measurable ...