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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1answer
33 views
For every monoid $M$ with zero is there a group $G$ such that $\mathrm{Grp}(G,G)\cong M$?
All monoids that I will consider here have identities. A monoid $M$ is said to have a zero iff $\exists z\in M \forall x\in M (zx=xz=z)$.
Let $M$ be any monoid with a zero. Must there exist a group ...
-1
votes
2answers
74 views
If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant? [closed]
If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?
4
votes
2answers
32 views
Regular $T_2$ space which is not completely regular.
Theorem 10. of
Pontryagin's Topological Groups says that:
Every Hausdorff topological group is completely regular.
But is there exists a Regular $T_2$ space which is not completely regular?
2
votes
1answer
40 views
Looking for example of an order homomorphism that doesn't preserve joins.
I know that not every order homomorphism preserves joins. But, I can't think of an example!
Both minimal examples and 'natural' examples welcome.
0
votes
1answer
26 views
Sufficient condition for reducibility of polynomial $f(x,y)$
[Dual to
this question]
Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied:
1) For every $x_0\in\mathbb{Q}$, the polynomial ...
2
votes
1answer
39 views
Sufficient condition for irreducibility of polynomial $f(x,y)$
Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied:
1) For every $x_0\in\mathbb{Q}$, the polynomial $f(x_0, y)\in\mathbb{Q}[y]$ ...
4
votes
0answers
38 views
An interesting space inspired by Mrowka's space
The example is from Bell M G. First countable pseudocompactifications[J]. Topology and its Applications, 1985, 21(2): 159-166..
Let us recall some necessary definitions firstly:
Let $X$ be a ...
1
vote
1answer
24 views
How to directly show that Figure 8 injective immersion is not a monomorphism
I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an ...
4
votes
0answers
53 views
Is $X$ pseudocompact
The following example with a little modified from the handbook of set theoretic topology, Page 574:
Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
3
votes
1answer
60 views
A completely regular topological space which is $T_0$ but not $T_1$?
The question pretty much says it all. I need a completely regular (the definition not requiring $T_1$) topological space which is $T_0$ but not $T_1$. I've sifted through Counterexamples in Topology ...
0
votes
1answer
44 views
Unique nearest point in epsilon neighborhood of compact real manifold?
I have to proof the following assertion:
Let $X$ be a compact submanifold of $\mathbb{R}^n$ and $\mathcal{U}^\varepsilon=\{p\in\mathbb{R}^n\;:\; |p-q|<\varepsilon \text{ for some }q\in X\}$. Then ...
1
vote
2answers
42 views
example of homotopy which is not path homotopy
Can someone give me a simple, concrete example of a homotopy, which is not a path homotopy?
Let $f, f'$ be continuous maps from $X$ to $Y$, and let $F: X \times I\to Y$ a continuous map such ...
1
vote
3answers
56 views
About the Pigeonhole principle
The principle says that: Let $k$ and $n$ be any two positive integers. If at least $kn+1$ objects are distributed among $n$ boxes, then one of the boxes must containat least $k+1$ objects. In ...
3
votes
2answers
57 views
Metric spaces and distance functions.
I need to provide an example of a space of points X and a distance function d, such that the following properties hold:
X has a countable dense subset
X is uncountably infinite and has only one ...
3
votes
2answers
79 views
Making Tychonoff Corkscrew in Counterexamples in Topology rigorous?
I'm reading pages 109 and 110 of Seebach and Steen's Counterexamples in Topology (p. 61 here) and I don't understand one of their steps. In particular, at the bottom of page 109 they say, "by ...
8
votes
1answer
115 views
A natural example in category theory
I'm looking for a natural example of a category $\mathcal{C}$ with finite limits (or just finite products) wherein some object $X$ is not isomorphic to a subobject of an inhabited object. In other ...
4
votes
2answers
56 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 ...
2
votes
1answer
41 views
Basically disconnected space which is not extremally disconnected
Space $X$ is basically disconnected if every cozero-set has an open closure.
Every extremally disconnected space is basically disconnected But i think the converse fails. The one-point ...
6
votes
3answers
85 views
What questions become answerable/computable given an uncountable character set?
Having reached the concluding portion of my first course in real analysis, one subject that I feel was not adequately addressed was the issue of cardinalities.
This is a subject I was interested in ...
11
votes
3answers
99 views
Are there interesting rings without unity?
There are several introductory textbooks which define a ring without any reference to a unity. However, nearly all of the rings one encounters in various branches of mathematics are endowed with a ...
0
votes
0answers
31 views
Repetitive tiling implies finite local complexity
My question probably needs to include the definitions of the terms in the title so I will first ask the question and then introduce the necessary definitions.
The following Theorem is stated without ...
2
votes
3answers
120 views
Example of topological space $(X,\tau)$ is Hausdorff but not discrete.
Would you give me an example of topological space $(X,\tau)$ such that $X$ is enumerable, and $\tau$ is Hausdorff but not discrete. How can I construct this?
2
votes
1answer
130 views
Infinite series involving $\sqrt{n}$
I am looking for examples of infinite series, whose sum is expressed as distributions or known functions, with a $\sqrt{n}$ in each term, such as:
$$ \sum_{n=0}^{\infty} \sqrt{n} z^n, \quad ...
2
votes
2answers
51 views
Does separability implies Lindelöf property?
Does separability implies a sort of Lindelöf property?
Since I can't prove this fact I'm beginning to think my conjecture is false.
Intuitively $\mathbb{R}$ has a countable subset $\mathbb{Q}$ which ...
1
vote
1answer
37 views
Show that floor function does't satisfy FTC.
The function is $f(x) = \lfloor 1-x^2 \rfloor$.
$$f(x) = \left \{
\begin{array}{lr}
-3 & : x \in [-2,-\sqrt{3})\\
-2 & : x \in [-\sqrt{3},-\sqrt{2})\\
-1 & : x ...
5
votes
2answers
123 views
Analysis without algebra
I once heard someone say that analysis is $99 \%$ algebra. He was, of course, referring to the amount of algebraic manipulations in the exercises from any calculus course.
I know that in topology, ...
1
vote
1answer
85 views
Derivative of complex-valued function and partial derivatives.
Let $f(x+iy)=u(x,y)+i\,v(x,y)$
Cauchy-Riemann Equations are satisfied at $z_0$
$u, v, u_x, u_y, v_x, v_y$ are defined on some open neighbourhood of $z_0$
$u, v, u_x, u_y, v_x, v_y$ are continuous at ...
5
votes
3answers
77 views
Use of infinity as an “idealistic approximation”
There have been several recent posts about the work of N. J. Wildberger, a finitist who seems to think that mathematics should only focus on things that have some sort of "real world" connection, ...
0
votes
1answer
62 views
Lim sequence $\neq$ lim subsequence
Let $\{x_n\}$ be a sequence and $\{y_k\}$ be a subsequence of $\{x_n\}$. Can you show me an example of $\{x_n\}$ such that $\displaystyle \lim_{k \to +\infty} y_k= +\infty$, but $\displaystyle \lim_{n ...
1
vote
1answer
74 views
There exists a continuous function that satisfies this property
Let $X$ be a non-compact subset of $\mathbb{R}$.
I want to show that there a continuous function $f: X \to \mathbb{R}$ such that $f$ is bounded but does not attain its bounds.
I think that there ...
1
vote
2answers
27 views
Examples of $F$-algebra which has no proper two-sided ideals but has one-sided ideal(s)
Let $F$ be a field and $A$ an $F$-algebra. (And assume that $A$ is finite dimension over $F$ if necessary.) A textbook says that $A$ is simple if it has no proper two-sided ideals.
To understand this ...
3
votes
0answers
40 views
Tensor products over field do not commute with inverse limits?
In the question: Inverse limit of modules and tensor product, Matt E gives an example where inverse limits and tensor products do not commute over the base ring $\mathbb{Z}$. He then goes on to show ...
1
vote
2answers
49 views
Number theory proof by counter example
Give an example of two cycles of lengths $r$ and $s$ respectively whose product does not have order $lcm(r,s)$
7
votes
2answers
188 views
Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$
Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$.
I don't really even know where ...
0
votes
1answer
40 views
Lagrange theorem for finite algebraic structures.
Let $S$ be a finite semigroupoid and let $a\in S$. The minimum of
$$\{n\in \Bbb N \mid a^{n+1}=a \}$$
, if it exists, is called the order of $a$ and is denoted by $o(a)$.
Which conditions on $S$, ...
2
votes
1answer
35 views
In a group, a finite left translation of $B$ covers $A$. Does any finite right translation of $B$ cover $A$?
Let $G$ be a group and $A,B\subseteq G$. Suppose there's some finite set $F\subseteq G$ such that:
$$A\subseteq FB$$
Is there any finite set $F'\subseteq G$ such that
$$A\subseteq BF'$$
?
0
votes
4answers
93 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 ...
103
votes
18answers
8k 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 ...
1
vote
1answer
50 views
Invariant Subspace Counterexample
Can someone give an example:
Suppose $T \in L(V)$. If $V = W \bigoplus W'$ and if $W$ is T-invariant then $W'$ is not necessarily T-invariant.
3
votes
0answers
31 views
Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?
Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?
Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.
...
3
votes
1answer
80 views
Worst category with first isomorphism?
I am no expert in category theory, but from VIII of Algebra: Chapter 0 I learnt that
In an abelian category every $A\xrightarrow{\phi}B$ can be decomposed into \begin{equation}A\twoheadrightarrow ...
4
votes
1answer
97 views
Convergence Counterexamples
I'm trying to compile a list of counterexamples for convergence implications (or rather, the lack of). I have an incomplete list and I hope to get it all together in one piece.
I'm currently working ...
3
votes
2answers
45 views
Structures where the injectivity of morphisms is forced
Can someone give me some examples of mathematical structures where the associated morphisms are forced to be injective (e.g. fields)?
Thanks
1
vote
1answer
42 views
Measure, absolutely continuous on boundary
Let $\mu$ be a finite nonnegative Borel measure on $\mathbb R^2_+=[0,+\infty) \times [0,+\infty)$ such that $\mu( \partial \mathbb R^2_+)=0$, i.e. $\mu$ is absolutely continuous on boundary. Is it ...
1
vote
1answer
46 views
If $f_n$ converges uniformly to $f$ on a measure space, show integral of $f_n$ converges to integral of $f$.
Please help me with this problem!
Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$.
If ...
4
votes
0answers
87 views
Separately continuous functions that are discontinuous at every point
What are some good examples of separately continuous functions $f: X \times Y \rightarrow Z$ that are discontinuous at every point?
Here's a theorem to rule out some spaces: link for a reference
...
3
votes
2answers
90 views
Counterexample in propositional logic
There is this lemma: Let $\Sigma\subset \textrm{Prop}(A)$ and $p, q \in \textrm{Prop}(A)$. Then $\Sigma\models p \implies \Sigma\models p\vee q$. I can't figure out a counterexample for the opposite ...
8
votes
6answers
285 views
Give an example of a simply ordered set without the least upper bound property.
In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact."
(The LUB ...
3
votes
4answers
56 views
Integral domains with non-trivial group of units that are not fields
I'm looking for examples of integral domains that are not fields but at the same time have more units than just the multiplicative identity 1.
It's clear to me that by Wedderburn's little theorem, ...
9
votes
1answer
115 views
Universal property characterizing $\Bbb R$
Is it possible to characterize the field of real numbers in a natural way with the language of category theory?
For example, $\Bbb Q$ is the initial object in the category of ordered fields and $\Bbb ...
