Tagged Questions

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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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. ...
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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 ...
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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 ...
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Example of a sequence with at least 3 limit points [closed]

What is an example of a sequence that has at least 3 limit points?
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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 ...
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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 ...
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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 ...
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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 ...
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What's the biggest number used as a counterexemple? [duplicate]

I'm looking for exemples of big numbers that are counterexemple of some interesting conjecture. Do you know conjectures that seemed to be true until a million (or many more) numbers where checked?
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counterexample relating to l'Hopital's rule

Suppose there are two funtions $f(x),g(x)$ such that (as $x \to a$) we have $f(x) \to +\infty$, $g(x) \to +\infty$, and $f'(x)/g'(x) = g(x)/f(x)$. Then by l'Hopital's rule, if $\lim f(x)/g(x)$ exists,...
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Connectedness and path connectedness of a union

Exercise: Let $A = \{(x, \sin (1/x)): 0<x\le 1\}$ and $B = \{(x,y)\in\mathbb R_{\le 0}\times\mathbb R | 0.5\le |y|\}$ be sets and $X = A\cup B$ the union. Show that $X$ is connected and path ...
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Open and connected set in metric space [duplicate]

In a normed space, we know that if a set is open and connected, it is path connected. Is it true for general metric space or general topological space?
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Is every Hausdorff homogeneous space also regular?

Every Hausdorff topological group is regular (completely regular, in fact). Is this true if I replace topological group with homogeneous space? This is not obvious to me because there are Hausdorff ...
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Sequence in $l^p$ but not $l^q$ for all $q<p$

I need to find a sequence for real $p>1$ so it is in $l^p$ but not in any of the space $l^q$ with $1 \leq q <p$. I tried the sequence $(1/n)^{1/q}$ which is in $l^p\setminus l^q$. However, this ...
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Function with infinitely many right inverses?

I was thinking about a function with infinitely many right inverses but I could not come up with anything. Does there exist a function with infinitely many right inverses?
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Suppose $d$ is a common divisor of two numbers $a_0$ and $a_1$, can $d$ be written as a linear combination of $a_0$,$a_1$ with integer coefficients?

Suppose $d$ is a common divisor of two numbers $a_0$ and $a_1$, can $d$ be written as a linear combination of $a_0$,$a_1$ with integer coefficients? i.e. there exists two integers $i_0,i_1 \in \Bbb Z$ ...
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Construct an example where x(t, x_0) is bounded but limt→+∞ x(t, x_0) does not exist.

Suppose we are given an IVP $x = f(x), x(0) = x_0$, x ∈ R^n for which we know that the unique solution x(t, x_0) exists globally in time. Construct an example where x(t, x_0) is bounded but limt→+∞ ...
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Does there exist a Hausdorff group which is not locally compact?

A topological space is countably compact if every countable open cover has a finite subcover. A topological space $X$ is locally compact if any point has a neighbourhood which is compact. A ...
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Examples and counter-examples for rings

Here's what I am trying to do: Listing mnemonics used: $D_n(\Bbb R)$ to denote diagonal $n \times n$ matrices with real coefficients. I.D. - Integral domain, E.D. Euclidean Domain Much of the ...