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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Sequence of continuous fuctions $f_n:[0,1]\rightarrow [0,1]$ s.t. $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ but…

Give an example of a sequence of continuous functions $f_n:[0,1]\rightarrow [0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ for every $\varepsilon >0$ but ...
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Bidual of a WSC space

Let $E$ be a Banach space which is weakly sequentially complete (i.e. each weak Cauchy sequence converges weakly). Must $E^{**}$ be weakly sequentially complete either? Of course, this question is ...
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Counterexample of Existence of a continuous extension of a Continuous function

Till now, I have proved followings; Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then, $Y=\mathbb{R}^k \Rightarrow \exists$ a continuous ...
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Is there an easy example of a vector space which can not be endowed with the structure of a Banach space

Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?
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Analytic Function with positive integers as zeros?

Do you know any nontrivial analytic function f(z) with zeros only at positive integer values of the argument z = 1, 2, 3, 4, ... ? If yes, please give some example. PS: I already thought of ...
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How to show that linear span in $C[0,1]$ need not be closed [duplicate]

Possible Duplicate: Non-closed subspace of a Banach space Let $X$ be an infinite dimensional normed space over $\mathbb{R}$. I want to find a set of vectors $(x_k)$ such that the linear ...
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Alternate definition for boundedness in a TVS

Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if for any open neighborhood $N$ of $0$ there is a number $\lambda>0$ ...
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Non-unital rings: a few examples

Every ring I've ever heard of is unital, i. e., contains a (unique) element $a$ such that $xa = ax = x$ for every $x$ in it. However, some rings do not have such an element. What are they? P. S.: one ...
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Noncompact sequentially compact space

Have you an example of a noncompact sequentially compact space, without using ordinal?
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1answer
397 views

Finitely Generated Group

Let be $G$ finitely generated; My question is: Does always exist $H\leq G,H\not=G$ with finite index? Of course if G is finite it is true. But $G$ is infinite?
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How to cook up test functions?

Let $\Omega\subset \mathbb{R}$ be open. A test function is a $C^\infty$ function with compact support. This is a rather strong restriction, for instance, no analytic function is a test function. But ...
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Poset Infimum and Supremum

I was asked to show that if every subset of a poset has an infimum then every such subset has a supremum. I did my proof and now I realize that what I was calling "infimum" was actually "a smallest ...
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Is this AM/GM refinement correct or not?

In Chap 1.22 of their book Mathematical Inequalities, Cerone and Dragomir prove the following interesting inequality. Let $A_n(p,x)$ and $G_n(p,x)$ denote resp. the weighted arithmetic and the ...
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1answer
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Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?

Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation} ...
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Examples of partial functions outside recursive function theory?

My math background is very narrow. I've mostly read logic, recursive function theory, and set theory. In recursive function theory one studies partial functions on the set of natural numbers. Are ...
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Providing a counter example for a Logic Statement

How do I give a counter-example of the following logic statement (I think the statement is false): There exists $x$ $\geq$ 0 s.t. (For All real $y$, $x$ = $y$$^2$) Since the statement has a "There ...
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201 views

Question about the cardinality of a space

I've been having conflicting thoughts about the following problem, and I was wondering if anyone could help me out. Is is true that the cardinality of every regular separable space does not ...
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484 views

nonstandard example of smooth function which fails to be analytic on $\mathbb{R}$

When I teach second-semester calculus I usually discuss the function $f$ defined by $$ f(x)=e^{-1/x^2} $$ for $x \neq 0$ and $f(0)=0$. Or, almost the same example, $g$ defined by $$ g(x)=e^{-1/x^2} $$ ...
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counter-examples in measure theory and set topology

The boundary of a subset of Euclidean space has empty interior, and furthermore has Lebesgue measure zero.Well,this is generally not true,but I can't find an explicit counter-example right now. ...
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convolution of a function with itself equals itself

In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed. for (1), $f*f=f$ ...
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Is there any complex-valued $C^\infty$ function $f(z)$ coincide with $z^{3/2}$ infinitely many times?

I want a $C^\infty$ function $f:U\rightarrow\mathbb{C}$, where $U\subset\mathbb{C}$ is a neighborhood of $0$, such that there exists a sequence of points $z_n\in U-\{0\}$ and ...
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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 ...
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Example of application of the Uniform Boundedness Principle

I've been trying to come up with an easy example of an application of the uniform boundedness principle (or Banach-Steinhaus theorem). I was thinking of something like the following, which is ...
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When can we say that a theorem has been proven?

I'm taking a Data Structures and Algorithms course for a CS program. The introductory material was all mathematics, mostly a series of formulas that we are to remember. I can work through the formulas ...
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What are the requirements for separability inheritance

Suppose we have an arbitrary separable topological space $X$. What are some (possibly nonequivalent) minimal requirements to put on $X$ to ensure that every subspace of $X$ is separable? This is not ...
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1answer
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What does the $A(m)$ denote in Engelking's book?

Recently, I'm reading Engelking's book. In this book, it always uses as the example the space $A(m)$, see for instance the example 1.1.8, page 15. For this space, I'm not very clear, for the ...
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2answers
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A question on a quotient of Alexandroff's double segment space

Does anybody know the Alexandroff's double segment space? References would be very welcome. I will try to describe it here: Alexandroff's double segment space: Suppose $X = C_1 \cup C_2$, where ...
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1answer
377 views

Application of Baire category theorem in Moore plane

The proof that Moore plane is not normal I have read was using Cantor's nesting theorem. But I heard that it is also possible to use Baire category theorem to prove and I want to know how. So, as ...
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What is a metric for $\mathbb Q$ in the lower limit topology?

A useful source of counterexamples in topology is $\mathbb R_\ell$, the set $\mathbb R$ together with the lower limit topology generated by half-open intervals of the form $[a,b)$. For example this ...
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0answers
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Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
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Every compact subset must be closed?

This is an exercise from a topological book. In $T_1$ space, every compact subset must be closed? For any two compact subset, their intersection must be compact? Thanks for any help:)
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Applications of quadratic forms

It seems that a lot of great mathematicians spent quite a while of their time studying quadratic forms over $\mathbb{Z},\mathbb{Q},\mathbb{Q_p}$ etc. and there is indeed a vast and detailed theory of ...
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Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
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Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
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A counterexample in topology

Semi-local simple connectedness is a property that arises in Algebraic Topology in the study of covering spaces, namely, it is a necessary condition for the existence of the universal cover of a ...
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Can two topological spaces surject onto each other but not be homeomorphic?

Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ and $g:Y\rightarrow X$ be surjective continuous maps. Is it necessarily true that $X$ and $Y$ are homeomorphic? I feel like the answer to ...
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Nonamenable subgroups of the unitary group of the hyperfinite II_1 factor

The hyperfinite $II_1$ factor arises as the group von Neumann algebra of any infinite amenable group such that every conjugacy class but that of the identity has infinite cardinality. The unitary ...
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If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
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1answer
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Intersection of countable set of compact sets

I am asking whether a specific construction is a counterexample to Theorem 2.36 in Rudin's "Principles..." book (3rd Ed.), which reads, 2.36 Theorem If $\{K_{\alpha}\}$ is a collection of compact ...
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1answer
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Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?

This question is related to the one here: A question related to krull akizuki In the answers to that question, some examples are given of a discrete valuation ring $A$ and a finite (necessarily ...
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A question about weakening the conditions of Schauder's fixed point theorem

I'm currently doing a course on the theory of metric spaces. This is the version of Schauder fixed point theorem from my course: Let $(X,\|\cdot\|)$ be Banach and $C\subset X$ a closed, bounded, ...
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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 ...
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2answers
346 views

predicate logic - counterexample $(A \models \phi \implies A \models \psi) \implies A \models \phi \rightarrow \psi$

It's predicate logic and I need to find a counterexample to disprove the follwowing claim $(A \models \phi \implies A \models \psi) \implies A \models \phi \rightarrow \psi$
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Deducing results in linear algebra from results in commutative algebra

Here are two examples of results which can be deduced from commutative algebra: Any $n\times n$ complex matrix is conjugate to a Jordan canonical matrix (can be proven using the structure theorem ...
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Does separability follow from weak-* sequential separability of dual space?

Let $E$ be a Banach space. Suppose that $E'$ is weakly-* sequentially separable, that is, that there exists a countable $D \subset E'$ s.t. every $x' \in E'$ is a limit point of a ...
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On nonconvex cones over compact convex sets in Hadamard spaces

Discussion http://mathoverflow.net/questions/6627/convex-hull-in-cat0 indicates the convex hull of a finite set can fail to be closed in a complete Hadamard space. Hence the following question should ...
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the meaning of “finite” in the finite covering theorem

A textbook I am using to learn analysis states (in reference to just the real line): Every system of open intervals covering a closed interval contains a finite subsystem that covers the closed ...
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1answer
286 views

Prove Axiom $10$ (Vector Spaces) independent of the others [duplicate]

Possible Duplicate: Is it possible to construct a quasi-vectorial space without an identity element? In Apostol Multivariable Calculus, $1.5$ exercise $30 b$, he asks the reader to prove ...
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1answer
92 views

Looking for some simple topology spaces such that $nw(X)\le\omega$ and $|X|>2^\omega$

I believe there are some topology spaces which satisfying the network weight is less than $\omega$, and its cardinality is more than $2^\omega$ (not equal to $2^\omega$), even much larger. Network: ...