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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15
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6answers
4k views

Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
-1
votes
1answer
62 views

Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...
4
votes
1answer
36 views

Definite integrals that are hard using the FTC but doable from first principles

When I was introduced to integration, I was briefly exposed to the definition of a Riemann sum, and as an illustration we worked out a couple of definite integrals directly from this definition: $$ ...
2
votes
1answer
34 views

Looking for example $\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f \,d\lambda$ where $\mu(A) < \infty$

I'm looking for a summable non-negative function $f: \Bbb{R} \to [0,\infty)$ and a measurable set $A$ with finite measure such that $$\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f ...
8
votes
1answer
106 views

Counterexample of polynomials in infinite dimensional Banach spaces

I'm trying to prove exercise I.3.B in Mujica's "Complex analysis in Banach spaces". DEFINITIONS: A map $P$ is an m-homogeneous polynomial from $E$ to $F$ if there is a m-linear map $A$ from ...
0
votes
1answer
22 views

Show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}})$ is in Sobolev space $W^{1,p}(B_1(0))$

As part of my seminar this semester, I need to show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}}) \in W^{1,p}(B_1(0))$. I have shown that $f$ is indeed in $L^p$, but could use some help proving ...
2
votes
1answer
46 views

Product of RREF versus RREF of product

Let $A$ and $B$ be two matrices of arbitrary shape where the number of columns of $A$ is the same as the number of rows of $B$. Is it always true that $$\textbf{rref}[A]\cdot \textbf{rref}[B] = ...
9
votes
1answer
92 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R} ^2$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
3
votes
1answer
159 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
3
votes
2answers
498 views

Example for a sequence of continuous functions which converge to a continuous function pointwise but not uniformly on a compact set.

My actual aim is to verify that each of the conditions in Dini's theorem is essential or not.Theorem says that A sequence {$f_{n}$} of continuous functions defined on a compact set $E$ converges to ...
3
votes
1answer
30 views

Example of a function which demonstrates this equivalent condition for continuity

Please note that I am not asking for a proof, just a confirmation of my understanding or a counterexample to the question I posed. We know that a function $f:M\to N$ for $M, N$ metric spaces is ...
5
votes
1answer
143 views

Smash products of pointed spaces is really not associative

The canonical bijective map $\mathbb{N} \wedge (\mathbb{Q} \wedge \mathbb{Q}) \to (\mathbb{N} \wedge \mathbb{Q}) \wedge \mathbb{Q}$ is not an isomorphism of pointed spaces (i.e. homeomorphism), see ...
27
votes
7answers
554 views

Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist?
4
votes
1answer
756 views

Totally disconnected implies base of closed sets?

Any $\ T_0$ space that has a base consisting of closed (hence clopen) sets is totally disconnected. Does a totally disconnected space necessarily have a base consisting of closed sets?
10
votes
2answers
178 views

Open set containing rationals but complement non-denumerable

I am taking Real Analysis classes and I got a homework that asks me: Give an example of an open set $\mathcal{A}$ such that $\mathcal{A}\supset\mathbb{Q}$ but $\mathbb{R}-\mathcal{A}$ is ...
1
vote
2answers
113 views

Example that differentiable functions do not form an integral domain?

Let $G$ be an open connected subset of $\mathbb{C}$ and $f,g$ be holomorphic functions on $G$ such that $fg=0$. If neither $f$ nor $g$ is zero over $G$, since $f$ and $g$ have countably many zeros ...
2
votes
2answers
26 views

Examples of product of two $L^1_\text{loc}$ functions that is not $L^1_{\text{loc}}$

Let $f\in L^1_\text{loc}$ and $g\in L^1_\text{loc}$, does $fg \in L^1_\text{loc}$? My textbook says it isn't in a general case. However if $g\in \mathcal{E} = \mathscr{C}^\infty$, then $fg\in ...
8
votes
2answers
333 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 ...
1
vote
1answer
22 views

Unbounded operators: product of adjoints strictly extended by the adjoint of product

It is well known that, if $T,S$ and $ST$ are densely defined operators on a Hilbert space $H$, then $T^* S^* \subset (ST)^*$. The proof of this is easy. Moreover, it's readily seen that equality ...
2
votes
1answer
41 views

Easy examples of the Arzela-Ascoli Theorem

Let $X$ be a compact metric space. $M \subseteq C(X)$ is relatively compact if and only if $M$ (i.e. its elements) is equicontinuous and uniformly bounded. I've been told that this theorem gives ...
29
votes
4answers
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 ...
6
votes
1answer
156 views

A metric on $\mathbb{R}^n$ such that $d(\lambda x, \lambda y)=|\lambda| d(x,y)$ which is not induced by a norm

Let $V=\mathbb{R}^n$. Let $d:V \times V\rightarrow \mathbb{R}$ a metric on $\mathbb{R}^n$. Assume that for any $x,y\in V$ and $\lambda \in \mathbb{R}$, we have $d(\lambda x, \lambda y) = ...
41
votes
14answers
4k views

An easy example of a non-constructive proof without an obvious “fix”?

I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...
5
votes
2answers
1k views

Why does the “zig-zag comb” weakly deformation retract onto a point?

I am starting to read Hatcher's book on Algebraic Topology, and I am a little stuck with exercise 6 in Chapter 0. Let $Z$ be the zigzag subspace of $Y$ homeomorphic to $\mathbb{R}$ indicated by ...
8
votes
1answer
1k views

Why doesn't the “zig-zag” comb deformation retract onto a point, even though it's contractible?

I am starting to read Hatcher's book on Algebraic Topology, and I am a little stuck with exercise 6(c) in Chapter $0$. Unfortunately a picture is involved so it doesn't quite make sense for me to ...
3
votes
1answer
86 views

Covering eleven dots in the plane with eleven coins - counterexample?

The questions here and here relate to the question as to whether, given ten equally sized coins and a configuration of ten dots in the plane, there is a way of placing the coins so that they cover all ...
0
votes
1answer
53 views

A bounded function having I.V.P. but not Riemann Integrable.

I am searching an example of a function $f$ on $[a,b]$ such that $f$ is a bounded function having intermediate value property but is not Riemann Integrable on $[a,b].$ Please give me such type ...
6
votes
3answers
1k views

Function whose image of every open interval is $(-\infty,\infty)$

How to find a function from reals to reals such that the image of every open interval is the whole of R? Is there one which maps rationals to rationals?
3
votes
3answers
66 views

Looking for pathologic counterexample: Nonzero harmonic function which is zero on the unit circle except 1

From my Complex Variables class: Let $C_1$ be the unit circle, $B_1$ the open unit disc and $\Gamma = C_1 \backslash \{1\} $. I am looking for a nonzero function $u \in C(B_1 \cup \Gamma)$ which is ...
6
votes
1answer
100 views

Finding a space with $X \cong X+2$ and $X \not\cong X+1$.

Question. Is there a topological space $X$ with $X \cong X+2$ and $X \not\cong X+1$? Here, $X+n$ denotes the disjoint union (i.e. coproduct) of $X$ with $n$ isolated points. This question is similar ...
0
votes
2answers
18 views

What is an example of a nonconstant subharmonic function that attains a minimum?

Let $D$ be a domain in $\mathbb{C}$. What would be an example of a nonconstant subharmonic function that attains its minimum in the domain?
1
vote
0answers
30 views

Positivity of this improper integral

I am not relieved of the following problem and trip. Let $\varepsilon>0$ be a small parameter, $a>0$ be a given constant, $x_{\varepsilon}\in(0,a)$ be a given sequence such that ...
2
votes
2answers
52 views

Normed algebra with bounded multiplication

Sometimes one finds the following definition of a normed algebra: This is an algebra with a norm on the underlying vector space such that there is a constant $K \geq 0$ such that $|x \cdot y| \leq K ...
2
votes
0answers
24 views

A question about a case where Central Limit Theorem doesn't apply

I'm trying to read Lehmann's "Elements of Large Sample Theory" and I have the following question about the text. The classical Central Limit Theorem is stated as: Now, the author goes on to ...
3
votes
1answer
55 views

Is the property “Existence of Antiderivatives” preserved under multiplication and composition?

Since differentiation is linear, we therefore have that if $f, g: I\to \mathbb{R}$ have antiderivatives (where $I\subset \mathbb{R}$ is an interval), then so does their linear combination. What if we ...
0
votes
1answer
104 views

non compact closed range operator

Lately I've been reading Abramovich and Aliprantis' book 'An invitation to operator theory', chapter 2 (page 69) on bounded below operators. I would like to find an example of non-compact (and ...
117
votes
0answers
3k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
1
vote
3answers
75 views

$f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs

Let $I \subset \mathbb{R}$ be an open interval and $f \in C^2(I,\mathbb{R})$. I am looking for an (simple) example of $f$ with the following properties ($x_0 \in I$) $f'$ is strictly monotonic ...
2
votes
2answers
7k views

Examples of types of mathematical models

I am a student currently doing a course on modelling and simulation. I came across the classifications of mathematical models and studied that they can classified as static or dynamic, deterministic ...
8
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1answer
291 views

Continuous functions and uncountable intersections with the x-axis

Let $f : \mathbb{R} \to \mathbb{R}$ such that the set $X = \{x \in \mathbb{R} : f(x) = 0\}$ does not contain any interval (i.e. there is no interval $I \subset X$) Of course the set $X$ can be ...
2
votes
1answer
32 views

Large deviation theory--examples of irregular sets

Let $(X,\tau)$ be a topological space, let $\mathcal{B}$ be its Borel $\sigma$-algebra, and $\mu_\epsilon$ be a family of probability measures on $(X,\mathcal{B})$. Suppose also that $\mu_\epsilon$ ...
2
votes
0answers
89 views

Game theory, Book by Tirole and Fudenberg, Never a weak best response,unclear example

In this book, I have the following problem: on page 446, there is a sentence: Note that $(0.9,0.9)$ is not removed by NWBR, as D is not dominated after C is deleted. I do not understand this "as". ...
3
votes
0answers
295 views

mid-point convex but not a.e. equal to a convex function

I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpinski's theorem from which we can deduce that for ...
18
votes
3answers
2k views

Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex

Rudin's Real and Complex Analysis Chapter 3 Exercise 4 is: Assume that $\varphi$ is a continuous real function on $(a,b)$ s.t. $$\varphi\left(\frac{x+y}{2}\right)\leq ...
7
votes
2answers
52 views

Non-measurable sets on $\mathbb{N}$

I'm familiar with the "construction" of non-measurable sets on $\mathbb{R}$. But of interest to me is if there is a way to construct a countably additive probability measure $\mu$ on $\mathbb{N}$ such ...
2
votes
2answers
680 views

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable. I know how to find the eigenvalues, and diagonalizing ...
2
votes
1answer
15 views

Eigenvalue alteration counter-proof

If $A$ is a square $n\times n$ matrix, with $\lambda_1,\ldots,\lambda_n$ being the eigenvalues of $A$, $v_1$ being the eigenvector associated with eigenvalue $\lambda_1$, and $d$ the column vector of ...
4
votes
1answer
262 views

Mid-point convexity does not imply convexity [duplicate]

A function $f: X \rightarrow \mathbb{R}$ is said to be mid-point convex if for all $x, y \in X$, we have $$f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2}. $$ Can you please give an example of a ...
2
votes
2answers
37 views

Constructing a (smooth) diffeomorphism between non-smooth manifolds

I've been trying to construct a smooth diffeomorphism between non-smooth manifolds. Unfortunately I don't think I know enough manifolds well enough to find an example of this. Mostly I've been ...
0
votes
1answer
17 views

How do I prove the Poisson integral formula for a harmonic function on exterior of a disk?

Let $R>0$ and $u:\mathbb{C}\setminus B(0,R)\rightarrow \mathbb{R}$ be a continuous function such that $u$ is harmonic on $\mathbb{C}\setminus\overline{B(0,R)}$. Assume that $u$ is bounded at ...