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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32
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5answers
6k views

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
3
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3answers
79 views

Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it? Though there are ...
1
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3answers
67 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
7
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3answers
5k views

Is a bounded and continuous function uniformly continuous?

$f\colon(-1,1)\rightarrow \mathbb{R}$ is bounded and continuous does it mean that $f$ is uniformly continuous? Well, $f(x)=x\sin(1/x)$ does the job for counterexample? Please help!
3
votes
2answers
208 views

An example of a bounded, continuous function on $(0,1)$ that is not uniformly continuous

I can not find the example of a continuous function on $(0,1)$ that is bounded on $(0,1)$, but not uniformly continuous on $(0,1)$. Is there any? Thank you.
4
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0answers
61 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
3
votes
5answers
284 views

How to find a differentiable function with bounded derivative satisfying some boundary conditions?

I am trying to find an example, preferably an explicit one, of a differentiable function $g:\mathbb{R}\rightarrow \mathbb{R}$ satisfying the following conditions: $\displaystyle g(0)=0, g(1)=1, ...
0
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1answer
152 views

Examples of decreasing sequences of closed sets with constant diameter and empty intersection in complete metric spaces

Looking through older exams from the topology class I'm taking, I found an interesting problem. Give an example: $ (X, d) $ - a complete metric space $ F_1 \subset F_2 \subset F_3 \subset ... $ - a ...
1
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1answer
36 views

An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and ...
1
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5answers
2k views

Counter-examples of homeomorphism

Briefly speaking, we know that a map $f$ between $2$ topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous. So, can anyone give me $2$ counter ...
3
votes
3answers
78 views

Example of Topological Group

I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology. ...
3
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1answer
51 views

Natural examples of multiplicatively-graded rings

A ring $A$ is multiplicatively graded if, as an additive group, it decomposes as $$A=\bigoplus_{n\geq 1} A_n$$ and the product satisfies $A_n \times A_m \rightarrow A_{nm}$. A natural example of ...
0
votes
0answers
14 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
3
votes
1answer
117 views

Examples of functions whose arc-length from the origin is given by their derivative

I'm looking for functions $y:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\int_{0}^{a} \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx = \frac{dy}{dx}\Bigg|_{a}$$ (this kind of feels like a ...
0
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0answers
28 views

Prove or Disprove If $T : R^n \rightarrow\ell_2$ is linear, then $T(A)$ is totally bounded in $\ell_2$ when $A ⊆ R^n$ is bounded

Prove or Disprove, If $ T: \mathbb R^n \rightarrow \ell_2 $ is linear, then It preserves total boundedness $ T(A) $ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded. I think ...
6
votes
1answer
472 views

Rainwater theorem, convergence of nets, initial topology

I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater's theorem. Let $X$ be a ...
12
votes
3answers
769 views

Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
7
votes
2answers
2k views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
0
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1answer
33 views

problems to understand a special definition of “free graded commutative algebra” from lecture

I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., ...
3
votes
2answers
255 views

Is there a first countable, 0-dimensonal, locally compact, lindelöf, non-compact space?

Is there a first countable, 0-dimensonal, locally compact, lindelöf, non-compact space in which all non-empty open sets have $\pi$-weight $\mathfrak c$? It also can be seen here. Thanks for your ...
0
votes
1answer
26 views

Why is semi-locally simply connected defined this way?

I would like to know why we define a space $X$ to be semi-locally simply connected if $\forall p\in X \exists U\ni p: i(\pi_1(U))=0\subset \pi_1(X)$ (SLSC), where $i$ is induced by ...
0
votes
1answer
54 views

Pseudocompact space having countable $\pi$-base but not metrizable

Can we find a normal Hausdorff space which is a countably compact locally connected space without isolated points and has a countable $\pi$-base but not metrizable? A collection $\mathcal{B}$ of ...
0
votes
1answer
35 views
1
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2answers
70 views

Double sequence, two sequences converge, but to different limits? [duplicate]

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, ...
0
votes
2answers
71 views

Example of a ring for which $rs \neq 0$ but $sr = 0$. [duplicate]

I am looking for an example of an associative noncommutative ring $R$ with the following property: for $r,s \in R$, $$ rs \neq 0, \text{ but } sr = 0. $$ Moreover, do rings for which this cannot ...
1
vote
0answers
65 views

Counterexamples in integral calculus: do functions like these exist?

Could you give me examples of functions of the following kinds? A function which is Riemann-integrable AND has an antiderivative, but is not continuous A function which is Riemann-integrable, and ...
0
votes
2answers
68 views

If $A\dot{-} B$ is countable and $B \dot{-} C$ is countable then $A\dot{-} C$ is countable? [closed]

Prove that: If $A\dot{-} B$ is countable and $B\dot{-} C$ is countable then $A \dot{-} C$ is countable? If not give a counter-argument
2
votes
1answer
51 views

Are there functions that are Holder continuous but whose variation is unbounded?

I have recently been introduced to the concept of Holder condition and I was told that there are functions that are Holder continuous but whose variation in unbounded. Can anyone present an example, ...
2
votes
2answers
109 views

Two sequences such that $\lim_{n \to \infty }| a_n-b_n |=1$ and only one of them is convergent

I should to find $a_n$ which converges to L (real number) and any sequence $b_n$ (which not converges to real number) so that they holds: $\lim_{n \to \infty }\left | a_n-b_n \right |=1$ Thanks for ...
7
votes
1answer
110 views

Does there exist a graph $G$ such that every edge is contained in a unique Hamiltonian circuit, that is not a cycle graph?

Suppose $G$ is an (undirected, simple) finite graph. If $G$ is a cycle graph, then each edge of $G$ belongs to a unique Hamiltonian circuit. Does there exist a non-cycle graph $G$ with this property?
0
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1answer
32 views

Does anybody know an example for a matrix with nullspace property for nonnegative signals?

A vector is $k$-sparse, if there are at most $k$ non-zero entries. In compressed sensing an arbitrary matrix $A\in\mathbb{R}^{m\times n}$ (with usually $m<n$) is said to have the null space ...
5
votes
3answers
475 views

Looking for a counter example for non-connected intersection of descending chain of closed connected sets

Let $X$ be a topological space and let $\left\{ Y_{i}\right\} _{i=1}^{\infty}$ be a descending chain of closed connected subsets of $X$. I know from reading elsewhere that ${\displaystyle ...
0
votes
0answers
28 views

False Counterexample for “for all sets A, B, and C, A ∩ (B - C) = (A ∩ B) - (A ∩ C)”

I've put together a proof on this, (which I would appreciate being verified), but I also want to know what a false counterexample might be for this? I'm new to discrete mathematics, and I'm honestly ...
1
vote
1answer
56 views

$\lim(a_n)\rightarrow 0$, $\lim(b_n)\rightarrow$ no limit, $a_nb_n\rightarrow 1$

Find a sequence $a_n$ such that its $\lim(a_n)\rightarrow 0$ and $b_n$ that $\lim(b_n)$ has no limit (finite or infinite) such that $\lim(a_nb_n)\rightarrow1$ using arithmetic i need to find a ...
0
votes
1answer
52 views

Limit point Compactness does not imply compactness counter-example

I think that I understand why compactness implies limit point compactness: Suppose $A \subseteq X$ has no limit points. Then $A^{\prime} \subseteq A$. Thus, $A$ is closed. Then for all $a \in A$, ...
5
votes
4answers
1k views

Example for a proper dense subspace?

I have been reading some books on functional analysis, and many of them keep talking about a vector space along with a dense proper subspace of it (especially when constructing counterexamples). But ...
5
votes
3answers
190 views

counterexample to a “theorem” on continuity of largest deltas for continuous functions $f:[a,b]\to\mathbb{R}$

"Theorem 12" in these notes states the following (verbatim): Let $f:[a,b]\to\mathbb{R}$ be continuous and let $\epsilon>0$. For $x\in[a,b]$, let ...
0
votes
1answer
24 views

Newton's Method in unconstrained optimization fails to converge

In order to show that Newton's method can produce a sequence of iterates that diverges, an example given in my book is apply Newton's Method to minimize $f(x)={2\over 3}|x|^{3\over 2}$. starting at ...
2
votes
1answer
47 views

What are some interesting cases where the two obvious definitions of “discrete object” diverge?

The nLab page defines "discrete object" as follows: Definition. [nLab] Let $\mathbf{C}$ denote a concrete category whose forgetful functor $U$ has a left adjoint $F$. Call the counit of this ...
1
vote
1answer
22 views

Equality of two multilinear forms

Take two multilinear forms $f,g$ defined on the same set $E$ such that $\forall x\in E,f(x,x,\dots,x)=g(x,x,\dots,x)$. Does that imply that the two functions are necessarily equal ? I can't seem to ...
-1
votes
1answer
39 views

Questions on symmetric difference of events

From a comment on my math overflow question: No, $P(A\bigtriangleup B)=0$ means $A$ and $B$ are essentially the same except in situations that almost surely do not happen. $P(A)=P(B)$ says much ...
29
votes
10answers
2k views

Non-associative commutative binary operation [duplicate]

Is there an example of a non-associative, commutative binary operation? What about a non-associative, commutative binary operation with identity and inverses? The only example of a non-associative ...
0
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2answers
44 views

strict local extremum of $f'$ that is neither saddle nor inflection value of $f$

Is there a function $f$ with the following properties: $x_0$ is a strict local extremum of $f'$. $(x_0,f(x_0))$ is neither a saddle point of $f$ (i.e. a point with $f'(x_0) =0$ which is not local ...
2
votes
2answers
164 views

Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
35
votes
6answers
3k views

Does commutativity imply Associativity?

Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples? ...
5
votes
5answers
526 views

Example of a relation that is reflexive but not symmetric

By definition, $R$, a relation in a set X, is reflexive if and only if $\forall x\in X$, $x\,R\,x$, and $R$ is symmetric if and only if $x\,R\,y\implies y\,R\,x$. I think $x\,R\,x$ can also be ...
4
votes
6answers
4k views

Non-associative, non-commutative binary operation with a identity

Can you give me few examples of binary operation that it is not associative, not commutative but has an identity element?
2
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1answer
165 views

A Lipschitz function with no directional derivatives at a point

I am stuck in constructing a function that is locally Lipschitz continuous at $x_0$ but it does not have directional differentiation at $x_0 $ in any direction. The definition of directional ...
0
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1answer
28 views

Can't the first (/second) transinfinite ordinal replace the first (/second) uncountable ordinal in several counterexamples?

An example of a sequentially compact but not compact space is $\omega_1$. Indeed, any sequence of countable ordinals either has infinite elements below some countable ordinal, or has an ascending ...
1
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2answers
102 views

Is there a well known nontrivial counterexample to this claim?

Suppose we have $A\subseteq\mathbb{N}$ with the property that if $B\subseteq\mathbb{N}$ and $B$ is finite, then $\exists a\in A\setminus\{ 1\}$, $\forall b\in B$, $\gcd(a,b)=1$. Are there any well ...