All Questions
Tagged with general-topology elementary-set-theory
1,228 questions
0
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0
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36
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Defining topological properties of functions without homeomorphisms
I started writing this thinking it was going to be more vague, but I ended up kind of half-attempting to answer my own question in the process and now my main question is whether or not my half-...
- 251
-4
votes
0
answers
88
views
Existence of a Bernstein set having the same cardinality as $\Bbb R$? [closed]
Does there exists a subset $S$ of $\mathbb{R}$ that is of the same cardinality as $\mathbb{R}$, such that neither $S$ nor $\Bbb R\setminus S$ contains any uncountable closed subset of $\mathbb{R}$?
I ...
- 169
1
vote
1
answer
66
views
Null law and Nullary intersection
I have been thinking of a problem that I could not get the most satisfying answer yet.
The problem I have been thinking of:
We know that the Null law says:
$$A\cap \varnothing = \varnothing$$
but why
...
- 2,311
0
votes
1
answer
48
views
How would I negate this mathematical statement (Basic Set Theory)?
Negate the following: For all $x\in G$, there exists a region $R$ such that $x\in R\subset G$.
I am struggling with how to interpret "$x\in R\subset G$". Is this the same as "$x\in R$ ...
1
vote
1
answer
202
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Question in Topology Without Tears by Sidney A. Morris.
I currently reading Topology Without Tears by Sidney A. Morris and I already read through eight chapters which topic is Finite Product. I try to show that "open interval (a,b) in $\mathbb{R}$ is $...
- 145
3
votes
1
answer
111
views
Symmetric correspondence between two sets
Consider two intervals $\displaystyle A,B\subseteq \mathbb{R}$. I want to define a correspondence $\displaystyle G:A\rightrightarrows B$ that satisfies the following two properties:
For any $\...
- 172
1
vote
1
answer
37
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Unclear part of Filter definition [duplicate]
In Bourbaki's General Topology, section on Filters, there is a part i find troublesome.
Here is a segment from the book:
6. Filters
1. Definition of a Filter
Definition 1. A filter on a set $ X $ is ...
- 1,671
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0
answers
71
views
How is this operation called and denoted?
$$\{(X_0,Y_0)\}\ast\{(X_1,Y_1)\} = \{(X_0,Y_0),(X_0,Y_1),(X_1,Y_0),(X_1,Y_1)\}.$$
How is the operation $\ast$ called? It is not Cartesian product but something similar.
If the operand sets of $\ast$ ...
- 5,197
1
vote
1
answer
83
views
Initial topology. Proof writing.
Let $X$ be a set. Let $(Y_i, \tau_i)$ be topological spaces and $f_i:X \to Y_i$ a family of mappings, $i \in I$. The set $X$ is endowed with the initial topology with respect to $\{f_i\}$. Let us ...
- 133
1
vote
0
answers
156
views
Compactness in $\mathbb{R^2}$ with subspace topology $\mathbb{Q^2}$ [closed]
We are considering $\mathbb{R^2}$ with euclidean topology with Subspace $\mathbb{Q^2}$.
We need to check whether the set $\{(x,y) \in \mathbb{Q^2}\mid \mathbb{x^2+y^2=1}\}$ is compact with subspace ...
- 43
0
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0
answers
41
views
Equivalence relation of adjunction space
Let $X$ and $Y$ be two topological spaces and $A \subset X$. Let $f : A \to X$ to be a continuous function. I constructed the adjunction space like here:
https://math.stackexchange.com/a/2032116/...
- 1,206
0
votes
1
answer
39
views
Product of closed sets is closed in product topology
I have this formula:
Let $X,Y$ two sets and condsider $C,E\subseteq X, \hspace{3pt} D,F\subseteq Y$.
It's true that $(C\times D)\setminus (E\times F)=\left((C\setminus E)\times D\right)\cup \left((C\...
- 1,497
1
vote
0
answers
85
views
On the intersection of nested sequence of nice sets [closed]
Let $\{H_n\}$ be a sequence of non-empty closed, bounded and convex sets in a Banach space $X$ with $H_{n+1}\subseteq H_n$ for all $n \in \mathbb{N}.$ Denote $\delta(A)= \sup\{\|x-y\|: x, y\in A\},$ ...
- 29
2
votes
0
answers
83
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Prove that the family of neighborhoods of $\Bbb N$ in $\Bbb R$ does not have a countable basis. [duplicate]
Let $X$ be a topological space and $A$ a nonempty subset of $X$. A subset $V$ of $X$ is called a neighborhood of $A$ if there exists an open subset $U$ of $X$ such that $A\subset U \subset V$. The set ...
- 2,543
4
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1
answer
103
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Are all $X \subseteq \mathbb{R}$ with $|(a, b) \cap X| = \mathfrak{c}$ and $|(a, b) \setminus X| = \mathfrak{c}$ homeomorphic?
Set $\mathfrak{c} = 2^{\aleph_0}$. Say that a set $X \subseteq \mathbb{R}$ has the property $P(X)$ if for any open interval $(a, b)$ we have $|(a, b) \cap X| = \mathfrak{c}$ and $|(a, b) \setminus X| =...
- 3,275
1
vote
1
answer
81
views
Can any non-empty open subset of $\mathbb R^n$ be written as a countable union of open balls whose centres belong to the open set itself?
Consider an arbitrary non-empty open set $\Omega \subset \mathbb R^n$. Is it true that one can find a countable collection of open balls $(B(x_l,r_l))_{l \in \mathbb N}$, where $x_l \in \Omega$ and $...
- 1,217
3
votes
1
answer
243
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Equivalent definitions of Cantor-Bendixson Rank
Yesterday I asked this question: Derived set of a closed subspace
The motivation of my question is that I am studying the notion of Cantor-Bendixson Rank, and I have found two different definitions. I ...
- 349
1
vote
1
answer
56
views
Does the existence of a countable basis on a metric topology imply existence of a countable collection of balls which is also a basis?
I think the answer should be yes, and I am trying to prove the positive answer.
Let $(X,T)$ be the topological space, and $G=\{ G_{\alpha} \}$ be our countable basis and $\mathbb{B}$ be the open ball ...
- 13.2k
1
vote
1
answer
77
views
Up to a null set, is the countable intersection of open sets $O_1,O_2,\dots\subset\mathbb R$ equal to a countable union of disjoint closed intervals?
Suppose $\{O_i\}_{i=1}^\infty$ is any sequence of open sets in the reals $\mathbb R$. Is is true that their intersection $\bigcap_{i=1}^\infty O_i$ is, up to a null set, equal to some at-most ...
- 5,869
0
votes
1
answer
85
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Show a unit sphere $\mathbb{S}^2$ and $\mathbb{R}$ are equivumerous
Let $\mathbb{S}^2$ denote the unit sphere in $\mathbb{R}^3$,i.e.
\begin{align*}
\mathbb{S}^2 := \{(x,y,z) \in \mathbb{R}^3 \vert x^2 + y^2 + z^2 = 1\}
\end{align*}
Show that $\mathbb{S}^2 $ and $ \...
- 79
4
votes
1
answer
90
views
Complete Boolean Algebra and Distributivity [duplicate]
Thear are two facts about Boolean algebra:
Every Boolean algebra is isomorphic to an algebra of sets. (Stone's Representation Thm)
Every complete algebra of sets is completely distributive. (said in ...
- 184
3
votes
4
answers
103
views
Why do we want topologies to be closed under both finite and infinite union but not infinite intersection? [duplicate]
So I recently read the definition of a topological space and a topology from a book, and according to it, the topology must be closed under finite and infinite union, but it must only be closed under ...
- 1,391
0
votes
0
answers
47
views
Showing a certain rule defines a topology
Let $L_1, L_2, L_3, \ldots$ be a family (sequence) of parallel lines in the plane, and let $X$ be an infinite union of $L_n$. Let us call a subset $G$ of $X$ big if either $G = \varnothing$ or $L_n \...
- 97
0
votes
1
answer
57
views
Show convergence of sets
Consider the following sets:
$$
\begin{aligned}
& A_n\equiv \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)\leq \delta_n\Big\}\\
& A \equiv \Big\{ x\in X: \lim_{n\rightarrow \infty} d\...
- 310
12
votes
1
answer
216
views
How many natural numbers are needed to generate 14 distinct subsets under complement and multiplicative closure?
Let the multiplicative closure operator $h$ be defined on $\mathcal{P}(\mathbb{N})$ by setting $hA$ equal to the smallest set containing $A$ that is closed under multiplication.
In L. F. Meyers' ...
- 1,563
4
votes
1
answer
259
views
Hard time grasping Dedekind cuts and real numbers [duplicate]
I have been trying to understand some elementary set theory recently, and am trying to understand how the real number line can be defined using the set of rational numbers. In particular, I am trying ...
- 528
1
vote
1
answer
179
views
Baby Rudin theorem 2.43: Why is it always possible to construct $V_n$ with $x_n \not\in \overline{V_{n+1}}$?
Code borrowed from here.
Let $P$ be a non-empty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable.
Here's the definition of a perfect set:
Let $(X,d)$ be a metric space, and let $P \subset X$. ...
1
vote
2
answers
85
views
Showing that the set of open sets of the finite-closed topology on $\mathbb{Z}$ is countably infinite
Here is what I have tried so far.
Define $F = \{X \subseteq \mathbb{Z} \ : \ |X| < \infty\}$ be the set of all finite subsets of $\mathbb{Z}$. Further define $F_k$ be the set of subsets of finite ...
- 361
2
votes
3
answers
149
views
Is it true that $A\in\{A\cup B\}$
I'm studying topology, and I've encountered a set $$\mathcal{T}_{\mathcal{B}} = \bigg\{\bigcup_{i\in I}B_i:B_i\in\mathcal{B}\bigg\},$$ where $\mathcal{B}$ is a basis for a topology, and $I$ is an ...
- 21
0
votes
2
answers
75
views
Internal points of $\mathbb{N}$
So, I have been taught that the empty set $\emptyset$ is both open and closed (if we can add "vacuously" it would be better).
$\emptyset$ is open because it contains all its internal points. ...
- 3,521
4
votes
2
answers
199
views
Topology Without Tears Exercise $1.1.9$ - Union of infinite rational open intervals
I'm working through the book Topology without Tears by Sidney A. Morris, and I've come to a roadblock while trying to solve exercise $1.1.9$, specifically with part iii. My background is primarily in ...
- 43
0
votes
1
answer
69
views
How do you prove that all countable, densely ordered sets without endpoints are isomorphic to the rationals?
I've looked online for a proof of this and have found several references to Canter's isomorphism theorem and the "back-and-forth method." However, I haven't been able to find any explicit ...
- 25
0
votes
0
answers
118
views
What is the union of sets that are elements of the empty set? [duplicate]
Let $(X,\tau)$ be a topological space with a base $\mathfrak{B}$. As $∅ ∈ \tau$, it must be possible: $∃ \beta \subseteq \mathfrak{B} | ∅=\bigcup_{G ∈ \beta}^{}G$. On the other hand, it can be: $\beta ...
- 79
0
votes
1
answer
76
views
Topology as an order (not order topology)
Given a topological space $(X,T)$, the topology $T$ is also a partial order with the inclusion relation $(T,\subseteq)$.
Given a continuous function $f:A\to B$ between two spaces $(A,T_1)$ and $(B, ...
- 111
2
votes
0
answers
61
views
Calculate the transitive interior and the transitive closure.
We say that a set $X$ is transitive whether for any $x$ in $X$ the implication
$$
(x\in X)\to(x\subseteq X)
$$
holds; moreover, we say that the set $S(y)$ is the successive of any set $y$ when the ...
- 5,110
13
votes
2
answers
222
views
Metric on the set of non-empty finite subsets (Ex. 2.4 MTH 427/527)
I am taking a course titled "Introduction to Topology I. General Topology" and stuck on the following exercise from the course notes:
Let $S$ be a set and $\mathcal F(S)$ denote the set of ...
- 133
0
votes
2
answers
40
views
Cannot understand this solution (max, min of a set)
$$A = \{ |x|: x^2+x < 2\}$$
Solving the inequality we get $-2 < x < 1$. Now in the notes there is written that this corresponds to write $A$ in this way:
$$A = [0, 2)$$
But I don't get how ...
- 3,521
0
votes
1
answer
38
views
Doubt on the infinite union of closed sets
I've been looking around (websites, notes and so on) but I never found a proof for this:
$$\bigcup_{n = 1}^{+\infty} \left[\frac{1}{n}, 1 - \frac{1}{n}\right] = (0, 1)$$
I am not understanding why ...
- 3,521
1
vote
2
answers
119
views
Sup and inf and boundary of this set
I have to find $\sup$, $\inf$ and $\partial A$ of the set
$$ A = \left\{ (-1)^n \frac{n^2+1}{2n^2+3}:\ n\in\mathbb{N} \right\}$$
So as $n\to +\infty$ we oscillate between $\pm \frac{1}{2}$. Those are ...
- 3,521
3
votes
1
answer
60
views
A set of mutually overlapping intervals
I need a reference for this elementary result which I assume to be true and known. Adding a citation in the paper I'm writing (even to a problem in a textbook) would be faster and more elegant than ...
- 33
4
votes
2
answers
77
views
Prove that in a topological space, empty intersection of a set and its boundary implies that the set is in the topology
As the title suggests, I would like to prove that in a topological space $(X,\tau)$ $$A\cap\partial A = \varnothing\implies A\in\tau$$ is true for $A\subset X$.
My attempt:
For every $x\in A,\,\...
- 43
0
votes
1
answer
136
views
Why require empty set in topology if any set contains empty set by definition
I don't look for exactly philosophical/motivational reason, i'm just confused because the almighty wikipedia says that $\forall A: \varnothing \subseteq A$, hence requiring empty set seems redundant (...
- 3
2
votes
1
answer
192
views
Derived set of subset $A$ of metric space $(X,d)$ is a closed set
I have to prove that derived set of subset $A$ of metric space $(X,d)$ is a closed set.
And here is my attempt: it is sufficient to prove $\overline{A^{'}}=A^{'}$, and it is equivalent to prove $(A^{'}...
- 101
2
votes
2
answers
126
views
If a metric space $M$ has only countably many open sets, prove that $M$ is countable
I found the following exercise
If a metric space $M$ has only countably many open sets, prove that $M$ is
countable.
in Kaplansky's Set theory and metric spaces (section 4.3, exercise 19)
I was ...
- 3,508
0
votes
2
answers
522
views
What is the difference between an improper subset and equal sets?
I've searched a lot but couldn't help myself with the distinction between an improper subset and equal sets. I would appreciate if someone could help me here.
- 123
0
votes
1
answer
110
views
Prove that $\mathrm{Cl}(\mathrm{Cl}(\mathrm{Cl}(\mathrm{Cl}(X)^c)^c)^c)^c=\mathrm{Cl}(\mathrm{Cl}(X)^c)^c$
$S$ is a metric space and $X \subset S$.
Define $\mathrm{Cl}(X)$ and $X^c$ as the closure and complement of $X$.
Prove that $\mathrm{Cl}(\mathrm{Cl}(\mathrm{Cl}(\mathrm{Cl}(X)^c)^c)^c)^c=\mathrm{Cl}(\...
- 308
1
vote
0
answers
65
views
Can the number of topologies on a set $X$ be equal to $2^{2^{|X|}}$? [duplicate]
Is there a non empty set $X$ such that $|T_X| = 2^{2^{|X|}}$ where $T_X$ are all topologies on $X$?
I know $T_X$ to be a strict subset of $\mathcal{P}(\mathcal{P}(X))$ and that no finite set works.
- 103
-1
votes
2
answers
52
views
A question about sup and inf
My professor last week told us about max, min, sup and inf of a set, and while talking about sup and inf, he stated that they always do exist. For example if $A = (2, 4)$ then $\sup (A) = 4$ and $\inf(...
- 3,521
6
votes
1
answer
71
views
Prove or disprove a particular statement in proximity spaces
A proximity space is a set $X$ with a relation $\sim$ between its subsets such that it satisfies the following axioms
$A\sim B\Rightarrow B\sim A$
$A\sim B\Rightarrow A,B\ne \varnothing$
$A\cap B\ne \...
- 175
1
vote
2
answers
205
views
Let $A,B,C \subset \mathbb{R}^n$ be convex, bounded and closed set, then $A+C=B+C \Rightarrow A=B$
Here a question and I would like to know if my solution is correct.
I think that it is not needed to use the fact that those sets are convexe, bounded & closed in order to solve the exercice, ...
- 825