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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3
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1answer
34 views

Zero-dimensional but not Hausdorff

Let's call a space zero-dimensional if it has a basis of clopen sets, and is $T_0$. Is there a zero-dimensional space that is not Hausdorff?
2
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4answers
517 views

How to come up with a counter example in linear algebra

This came up in a problem I was working on. Problem:Let $V$ be an $n$ dimensional vector space over a field $F$. Let $T:V\rightarrow V$ be a linear operator and let $W$ be a $T$ invariant subspace of ...
0
votes
2answers
66 views

Graph connected does not imply $f$ is continuous [on hold]

Show an example of a function $\newcommand{\R}{\mathbb{R}} f: \R \times \R\to \R$ such that $f$ is not continuous, but its graph $$ \Gamma_f := \left\{\bigl((x, y), f(x, y)\bigr) \mid \text{$(x, y)$ ...
0
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0answers
10 views
1
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0answers
38 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
12
votes
2answers
389 views

An example of a space which fails to be compactly generated

Does anyone know of an example of a topological space which is not compactly generated? I am using the definition in May's book "A Concise Course in Algebraic Topology." The definition is that a space ...
1
vote
1answer
31 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
1
vote
1answer
28 views

Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
183
votes
28answers
17k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
0
votes
1answer
36 views

Non-standard examples of Distributions

Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions. Additional: is there a ...
0
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1answer
12 views

What is an example of $E/F,L/E$ are normal but $L/F$ is not. [duplicate]

Let $E/F,L/E$ be normal field extensions. What would be an example such that $L/F$ is not normal?
1
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2answers
14 views

What is an example of transcendental extension such that a monomorphism cannot be extended?

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$. Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism. What is an ...
9
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9answers
824 views

Advice on finding counterexamples

I am reaching out for specific advice on how one should go about finding counterexamples. It seems almost every time I've ever attempted a "find a counterexample" problem, I have to cheat by asking a ...
11
votes
4answers
190 views

Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
1
vote
1answer
35 views

Example of a white noise series that is not a martingale difference series with respect to its natural filtration

For a homework exercise, I am asked to find an example of a white noise series that is not a martingale difference series with respect to its natural filtration. Does anyone know an example? I read ...
8
votes
3answers
229 views

A strong Hausdorff condition

Is the following strong form of Hausdorff equivalent to usual Hausdorff? $X$ is strong Hausdorff if given distinct elements $x,y$ in $X$ there are open sets $U,V \subseteq X$ with $x \subseteq ...
2
votes
0answers
24 views

How much regularity is needed, anyway?

When doing real analysis, the difference between functions which are continuous and functions which are not is intuitive. The graph of the later may exhibit shearing, or extreme distortion (in higher ...
4
votes
3answers
59 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
0
votes
1answer
21 views

Are there an completely regular, non-Lindelöf spaces with only constant real valued functions?

Is there an example for a topological space, which is not Lindelöf, but is completely regular, on which continuous real valued functions, are, constant for all $x \in X$, or, constant for all, except ...
0
votes
2answers
16 views

Example of $1-1$ Correspondence with Subgroups of Factor Group

I am working out an example to deomstrate the one-one correspondence between $\{\text{subgroups of}\ D_4/N\}$ and $\{\text{subgroups of $D_4$ that contain $N$}\}$ but I am short one in $D_4$. ...
2
votes
1answer
32 views

Finding a counter example for $ \left(A+A\right)'\subseteq\left(A'+A\right)\cup\left(A'+A'\right)$

let $A'$ be the set of limit points of $A\subseteq\mathbb{R}$ and $A+B=\{x+y:x\in A,y\in B\}$, I'm required to find a counter-example for: $$ ...
6
votes
2answers
119 views

If $\frac{a+b}2$ is rational, can we say that $a,b$ are rational?

The question is if it's given that $$ {a+b\over 2} \in \Bbb Q $$ prove or disprove $a,b \in \Bbb Q$. Since it is to disprove, I tried the following method by using examples. Take $$a = 1 + \sqrt{2} ...
6
votes
3answers
1k views

Reflexive space which is not uniformly convex

I found this beautiful theorem due to Milman and Pettis: Every uniformly convex Banach space is reflexive. I think it's a remarkable statement, since uniformly convexity is a geometric property ...
1
vote
1answer
16 views

Construct measures on $\sigma(B)$ that agree on $B$

Let $X=\{ 1,2,3,4\}$ and $\mathcal B=\{\{1,2 \},\{ 1,3\},\{ 2,4\},\{ 3,4\} \}$. And let $\mathscr A = \sigma(\mathcal B)$ be the $\sigma$-algebra generated by the set $\mathcal B$. I wish to construct ...
0
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1answer
47 views

Example for $a^k\equiv b^k$ and $k\equiv j$ but $a^j\not\equiv b^j\pmod n$

I need some help in the number theory please , Who can give me an example : If $$a^k≡b^k \pmod{n}$$ and $$k≡j \pmod{n}$$ is not necessary to be $$a^j≡b^j \pmod{n}$$
0
votes
1answer
10 views

example verification partial order

Let A= { 1, 2, 3} then R= { (1, 1), (2, 2) , (3, 3) , (1, 2), (2, 3) } ,the relation R is reflexive and anti-symmetric,i get it, but how is it following the transitive property for it to become ...
0
votes
1answer
19 views

Cavalieri's Principle in measure theory

The first part of Cavalieri's principle (in measure theory) states if $E$ is measurable, then almost every slice $E_x$ of $E$ is measurable. Here, it uses "almost every", so what is an example where ...
1
vote
1answer
953 views

Example of a Markov chain transition matrix that is not diagonalizable?

It is well-known that every detailed-balance Markov chain has a diagonalizable transition matrix. I am looking for an example of a Markov chain whose transition matrix is not diagonalizable. That is: ...
2
votes
4answers
62 views

An example of a space $X$ which doesn't embed in $\mathbb{R}^n$ for any $n$?

Apologies if this has been asked before, but couldn't find it. The definition of embedding that I'm using is this: Suppose $X$ and $Y$ are topological spaces. We call a function $f:X\rightarrow ...
1
vote
1answer
57 views

closed, convex, absorbing subset of a banach space

There is a nice theorem that every closed, convex, absorbing subset of a banach space includes an open ball arround $0$. Can you give an example where the theorem fails if we do not assume the subset ...
5
votes
1answer
93 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
5
votes
2answers
220 views

Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...
5
votes
1answer
50 views

McLain's characteristically simple group

Given a field $\mathbb{F}$, one can construct the free $\mathbb{F}$-module over the set of rational numbers. Assume that $(v_x)_{x \in \mathbb{Q}}$ is a basis of this linear space (indexed by the ...
7
votes
3answers
1k views

Need some help on a non-example of equicontinuity

In an attempt to better understand the definition of an equicontinuous family of continuous functions, I want to find a simple non-example. My intuition says that the family ...
165
votes
32answers
9k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
1
vote
1answer
17 views

Counterexample for continuous function over product topology without compactness

Suppose $f$ $(X,d_x)$: $\rightarrow$ $(Y,d_y)$ is a function between metric spaces, and $X \times Y$ has the product topology. The graph $G_f$ is the subspace $G_f$ = {$(x,f(x))$ | x $\in$ $X$}. If ...
1
vote
2answers
94 views

What would be an example such that $aH=bH$ but $Ha \neq Hb$?

Let $H$ be a subgroup of a group $G$. Assume $aH=bH$ for some $a,b\in G$ What would be an example such that $Hb\neq Ha$? I cannot imagine what would be.. Moreover, if $H$ is infinite, ...
-1
votes
0answers
33 views

If $\int_{c}^d g_x(y)\ dy $ is integrable, then is $g_x$ is integrable?

Let $g:[a,b]\times [c,d]\to \mathbb R$ and let $g_x:[c,d]\to \mathbb R$, $g_x(y)=g(x,y)$ where $x$ is fixed. If $\int_{c}^d g_x(y)\ dy:[a,b]\to \mathbb R$ is (Riemann) integrable does that necessary ...
2
votes
1answer
288 views

A semisimple commutative Banach algebra with a non-semisimple quotient

I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient. Attempt from the comments: "I take $A$ to be the algebra of all continuously ...
3
votes
3answers
124 views

Examples of quasigroups with no identity elements

If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general). What would be some examples of quasigroups without an identity ...
1
vote
1answer
26 views

What kind of functions lie in $L^2(\Omega )-H^{1/2}(\Omega )$?

It's well known that a discontinuous function does not lie in $H^{1/2}(\Omega )$ if $\Omega\subset\mathbb{R}$ is one dimension. My question is, For $\Omega\subset \mathbb{R}^3$, are there any simple ...
2
votes
0answers
32 views

Not continuous function with closed graph

I would like to see an explicit example of a function $f:R\rightarrow R$ which has a closed graph in $R^2$ but is discontinuous at every point in the real line.
1
vote
1answer
28 views

Closed Subsets of the Real Line that are Uncountable

If a subset of the real line is uncountable and closed, does it have to contain a closed interval? Is there any theorem related to this?
2
votes
1answer
32 views

Is this an example of a sequential non-Fréchet–Urysohn space?

Let $X$ be the set $X = \{ (0,0) \} \cup \{ (\frac{1}{n},0) : n \in \mathbb N \} \cup \{ (\frac{1}{i},\frac{1}{k}) : i,k \in \mathbb N \}$. Points of the form $(\frac{1}{i},\frac{1}{k})$ are ...
2
votes
1answer
82 views

Example of $\deg(fg)<\deg(f)+\deg(g)$

Let $R$ be an integral domain and $f,g\in R[X_1,...,X_n]$ where $n>1$. What is an example of a pair $f,g$ such that $\deg(fg)<\deg(f)+\deg(g)$? Moreover, i have proven that the units of ...
1
vote
0answers
44 views

specific magma examples

Give an example of a magma $S$ such that $S$ has a zero and $S$ has a left zero divisor that is not a right zero divisor an example of a magma with an identity such that there is an element with ...
0
votes
1answer
22 views

Oscillating essential discontinuities exist?

Let $f$ be a function $\mathbb R \to \mathbb R$. According to Wikipedia an discontinuity of $f$ is essential if and only if either the left or the right limit is infinite or does not exist. Is it ...
1
vote
1answer
39 views

Continuity and interior

I have questions about the relation between continuity and interior based on the article ;Continuity and Closure At first I guess that there will be a property like $f:X\rightarrow Y$ is continuous ...
4
votes
3answers
928 views

Basic examples of monoids?

What are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply because they ...
38
votes
9answers
1k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...