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Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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how to find $\text{int}(D)$

On the set of all real numbers with usual topology, let $ D = \bigcap_{n=1}^{\infty} \left(0, 5 + \frac{1}{n} \right)$ Find $\text{int}(D)$
hnuyq's user avatar
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Product of quotient map $p: X \to Y$ and identity $Id_Z$ is a quotient map when $Z$ is compact and Hausdorff

Let $X,Y,Z$ be topological spaces where $Z$ is compact and Hausdorff. Let $p: X \to Y$ be a quotient map. I want to show that $$p \times Id_z: X \times Z \to Y \times Z: (x,z) \mapsto (p(x),z)$$ is a ...
Robski's user avatar
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1 answer
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Construct a topology $T$ on $X=(-\infty,0]$, such that $X$ is compact? [closed]

Construct a topology $T$ on $X=(-\infty,0]$, which differ from trivial topology and discrete topology, such that $X$ is compact?
hnuyq's user avatar
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For all $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such that $a'\in A$ and $c'\in C$. What can we conclude about $A,C$?

The $\mathbb R^n$ space is partitioned into three sets $A,B,C$. Given: $B$ is convex. $A,C$ are non empty. For all open segment $(a,c)$ where $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such ...
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Algebraic Topology: Deformation Retraction and Quotient Spaces (using Mobius )

Could someone please help me with this question (and its solution)? Thanks!! Solution Part 1 (which shows that [0,1] x {1/2} deformation retract of [0,1] x [0,1]): I don't get how the homotopy gets ...
user1325970's user avatar
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partition of $\Bbb R^3$ by punctured topological surfaces

Consider a partition of $\Bbb R^3$ by topological surfaces $M=\Bbb R^2-\lbrace \mathrm{1~point} \rbrace$. Is it possible to make these surfaces $M$ complete, hence giving a partition of $\Bbb R^3$ ...
zeta space's user avatar
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topology effect on PDE solution

I'm interested in understanding how the topology of a domain influences the solutions of a PDE defined on that domain. Specifically, I'm curious if there are methods to explore different topological ...
InfSuplife's user avatar
-1 votes
0 answers
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how to build a topology T on X such that X is disconnected? [closed]

how to build a topology T on X=(0,5), which difference from trivial topology and discrete topology such that X is disconnected?
ivelyic's user avatar
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0 answers
28 views

continuity of a function defined in a compact metric space

Let $K$ be a compact metric space and $S\subset K$ a compact subspace such that $$S \subset \bigcup_{i=1}^{k} B_{s_i}$$ where each $s_i \in S$ and $B_{s_i}$ are open balls centered at $s_i$ consider ...
Victor's user avatar
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Point-separating fields of clopen sets on compact spaces without Choice

In Matthew Dirk's Paper on Stone's representation theorem there is a proof of Lemma 3.8. If X is a Stone space and F is a separating field of clopen subsets of X, then F is the dual algebra of X; that ...
Daniel Weichhart's user avatar
4 votes
2 answers
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Derived set of a closed subspace

Suppose $X$ is a compact Hausdorff topological space with a basis of clopen sets. Let $A$ be a closed subspace of $X$ and let $A^{(0)}=A$, $A^{(1)}=A^\prime$, etc. My question is: it is true that $A^{(...
Earnur's user avatar
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If there is a point in between that is also in set $X$, then $X$ is open. [closed]

Let $X$ and $Y$ be half spaces. Condition 1: For all $x\in X$ and $y\in Y$, there are points $x'$ and $y'$ between $xy$ such that $x'\in X$ and $y'\in Y$. Can condition 1 implies openness of $X$ ...
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1 vote
2 answers
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Topology: Torus and Quotient Spaces

just started with my first course in abstract mathematics, namely Point-Set Topology. Would appreciate some help here! ($I = [-1,1]$) I have a very simple question on a Torus. Firstly, I have seen ...
user1325970's user avatar
1 vote
1 answer
60 views

Embedding $S^1$ into Möbius strip

An assignment problem I'm working on right now is formulated as follows: Definition: The Möbius band is the quotient of the square $I^2 = [0,1]^2$ by the relation $(0,y) \sim (1,1-y)\forall y \in I$. ...
Jakob Lynas's user avatar
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Proof by induction of $V(r) \subset V(s)$, where the Vs are neighboords of $e$ in a topological group.

Some context: I'm trying to fill the details of a proof concerning neighborhoods indexed by dyadic rational numbers. Suppose we have a sequence $U_n$ of symmetric neighborhoods of $e \in G$ such that $...
none's user avatar
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0 votes
1 answer
95 views

A topology different from discrete and anti-discrete, in which every open set is closed, and every closed set is open. [closed]

To give an example of a topology on a set of four points, different from discrete and anti-discrete, in which every open set is closed, and every closed set is open.I have 0 ideas at all, I thought ...
Nitro Kansas's user avatar
-3 votes
0 answers
19 views

prove that every infinite subset of X is dense in X with respect to the cofinite topology [closed]

I know subset A of X is said to be dense in X, if A_bar = X. If F is a closed set in X containing A, then F = X ,How to i show that for every infinite subset also added it is a cofinite topology ...
Suraj Shaw's user avatar
2 votes
0 answers
46 views

Is metric incompleteness of $\overline{X}$ always witnessed by some sequence in $X$?

This following conjecture seems intuitively true to me, but I have trouble proving it (and I haven't found it confirmed/disconfirmed in the literature or online): Let $M$ be an incomplete metric space....
fr_'s user avatar
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0 answers
19 views

Is there a classification of connected sets in $\mathbb{R}^2$? [duplicate]

We know that in $\mathbb{R}$ with standard topology, a subset $A$ of $\mathbb{R}$ is connected if and only if $A$ is an interval (including points, rays, $\mathbb{R}$). Is there a classification of ...
with-forest's user avatar
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1 vote
1 answer
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Singletons in regular spaces

I need help with a statement. Let $X_1$ and $X_2$ be topological spaces and consider $X = X_1 \times X_2$. Suppose the product space $X$ is regular, that is, for any closed subset $C \subset X$ and ...
Joel Marques's user avatar
0 votes
1 answer
50 views

Give examples of open and closed sets

Task: Let the topology $T = \{\emptyset; \mathbb{R}; \left(-\infty, 1/2^{n}\right], n \in \mathbb{Z}\}$ be given on the line $X=\mathbb{R}.$ Give examples of: a) an open but not closed set b) a closed ...
susem's user avatar
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1 vote
1 answer
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A certain distance-related map is well-defined

I am trying to understand this answer. Context: we want to show that there exists a retract $C\to A$ where $C$ is the Cantor set and $A\subseteq C$ is a nonempty closed subset. They suggest to take a ...
runyoucleverboy's user avatar
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0 answers
40 views

Find a bounded real sequence, whose range has exactly one accumulation point, but which is not convergent [duplicate]

Example of bounded sequence on $ \mathbb{R}$ s.t. the set $\{X_n : n \in \mathbb{N}\}$ has exactly one accumulation point but $X_n$ is not convergent My thoughts: Since Xn is bounded it has ...
GaloisRocks's user avatar
0 votes
1 answer
42 views

Prove that open balls are preserved under scalar multiplication

Suppose we have a random ball $B_r(x)$ in $ \mathbb{R²}$, and we use the vector space structure and scale this ball down by a non zero scalar $a$. I believe that, $\frac{B_r(x)}{a}= B_{r'}(x')$. My ...
Babu's user avatar
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1 answer
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A closed subset of separable normed space $X$ is separable.

Let $(X,||\cdot||_X)$ be a normed separable space and let $F$ be a closed subset of $X$, then $F$ is separable. My attempt Let $D\subseteq X$ countable and dense, that is $cl_X(D)=X$, then $D\cap F\...
NatMath's user avatar
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What would fail if "second countable" is replaced by "paracompact" in the definition of manifold?

If I were to define a topological manifold to be a locally $n$-Euclidean, Hausdorff, paracompact topological space and continue to add the smooth structure based on it, what important result would ...
user108580's user avatar
1 vote
0 answers
47 views

Finite product topology and some properties

I need help with two items. First, I would like you to check if the proof I wrote for the following statement is correct: Let $\lbrace X_j \rbrace _{j=1}^n$ be a finite collection of topological ...
Joel Marques's user avatar
-3 votes
1 answer
39 views

A point in $ X$ is represented by $X\setminus \overline{\{x\}}$ [closed]

Let $X$ be a topological space, I read that $x$ can be represented by the set $\tilde x=X\setminus \overline{\{ x \} }$. Does this mean that the map $f: X \to O(X)$ sending $x$ to $\tilde x$ is a ...
Catalio13's user avatar
2 votes
1 answer
49 views

Proof that a normal ($T_4$) pseudocompact space is countably compact

I am having little trouble understanding the proof for this in Counterexamples in Topology (page 20). It is a proof by contradiction. Suppose you have a you have normal pseudo compact space $X$ which ...
riescharlison's user avatar
1 vote
1 answer
69 views

How does the size of $\{x: f(x) \text{ is locally constant}\}$ relate to the the size of $\text{im}(f)$?

I have been wondering (though I am not asking in this post) whether a function $f:\mathbb{R}\to\mathbb{R}$ is locally constant almost everywhere $\iff$ $\text{im}(f)$ is countable. where by "...
Sam's user avatar
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0 votes
1 answer
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Which all topological properties should be preserved for a map to be continous?

Following this post Topological properties preserved by continuous maps, we see that continous functions preserve only some topological properties. My question is, from these some, is there any ...
Babu's user avatar
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1 vote
1 answer
32 views

Continuity and uniform continuity on products

Let $X,Y,Z$ be metric spaces, and $F:X \times Y \rightarrow Z$ be a continuous function. Further, suppose that for each $x \in X, F_x: Y \rightarrow Z$ is uniformly continuous. Then, is the canonical ...
Anupam's user avatar
  • 434
4 votes
1 answer
60 views

Is the set of all antichains of $\omega^{<\omega}$ avoiding all chains Borel?

Let's identify $\mathcal P(\mathbb N^{<\mathbb N})$ with $2^{\mathbb N^{<\mathbb N}}$, so it is a Polish space. The set $\mathcal A=\{A\subseteq \omega^{<\omega}: A \text{ is a maximal ...
Pink lake's user avatar
1 vote
1 answer
51 views

A full embedding functor from $Sob$ to $Frm$?

In the following passage, the authors Picado and Pultr say the functor is a full embedding then they say that it would be a full embedding under some condition. Any explanation?
Catalio13's user avatar
0 votes
1 answer
27 views

Reconciling Continuity of Binary Relations with Continuity of Functions/Correspondences

I asked this question in the Economics StackExchange as well, but figured it may be better-suited here. There are various ways to express the concept of continuity of a binary relation, but one I've ...
hillard28's user avatar
-1 votes
0 answers
32 views

Prove that $U = K^{\mathrm{o}}$ for U and K compact in a non-compact space X. [closed]

Let $(X,\tau)$ be a non-compact Hausdorff-space and $\varnothing \neq U \in \tau$ with $\bar{U}$ (closure) compact. Let $K$ be a compact subset of $X$ where $K^{\mathrm{o}}$ (interior) and $X\...
Viggo De Boom's user avatar
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0 answers
17 views

Grassmannian manifold and subspace of transverse pairs

I am struggling with the following exercise. Consider the subspace $W\subseteq Gr_k(V)\times Gr_l(V)$ ($k+l = \dim(V)$) of pair $(A,B)$ s.t. $A$ is transverse to $B$. Prove that $W$ is open. My ...
Ronnie's user avatar
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1 vote
1 answer
37 views

An auto-homeomorphism of a grid of balls maps the center to itself.

Suppose you have 9 closed balls in $\mathbb{R}^3$, stacked into a 3x3 grid, as displayed in the (not perfect) drawing. s.t. the upper leftmost ball touches the North Pole of the ball below it ...
n-0's user avatar
  • 133
0 votes
1 answer
30 views

Why is a crossing of 2 arcs defined like this here?

The below text defines a crossing for 2 arcs, where $a_ib_j$ is a Jordan arc of finite length terminating at 2 distinct points $a_i$ and $b_j$: Two arcs $a_i b_j, a_k b_l(i \neq k, j \neq l)$ are ...
Princess Mia's user avatar
  • 2,497
0 votes
1 answer
41 views

Understanding the difference between limit point compact and compact

I have come across a counter example for showing a set that is limit point compact is not necessarily compact, however I cannot quite see how the example shows the result. The definitions I am working ...
JamesLevine's user avatar
1 vote
1 answer
57 views

Stone's Representation Theorem equivalences

I have seen statements such as "Stone's Representation Theorem is equivalent to Compactness, Tychonoff, Ultrafilter Lemma, Boolean Prime Ideal Theorem, Completeness Theorem.... over ZF" ...
Daniel Weichhart's user avatar
1 vote
0 answers
51 views

What is $\bigoplus_{n\in \mathbb{N}} I^{n}$, is topology is Tychonoff subspace topology?

I want to know what is $\bigoplus_{n\in \mathbb{N}} I^{n}$ with $I=[0,1]$. Im not sure if is just: $$\bigoplus_{n\in \mathbb{N}} I^{n}=\{(x_{n})\in \prod_{n\in \mathbb{N}}I^{n}\mid \exists ~M\...
Yves Stanislas SH's user avatar
1 vote
0 answers
26 views

The functor $Sob \to Loc $ is faithful

I want to show that the functor $O: Sob \to Loc$ from the sober topological spaces to the locales is faithful. So take $X,Y$ two sober topological spaces. I want to show that the map $$O_{X,Y}: Hom(X,...
Catalio13's user avatar
0 votes
0 answers
28 views

Question about the Vector Bundle Chart Lemma

In Lee's Introduction to smooth manifolds, one may find the following lemma: I was trying to construct a similar criterion for continuous vector bundles. Would we need condition (iii) for that? As ...
Julius Maximus's user avatar
7 votes
0 answers
219 views
+50

Topologies that make a map continuous

I was studying general topology when a question came to my mind. The definition of initial (final) topology induced by a map and a topology on its codomain (domain) is rather common. It's somewhat ...
Amanda Wealth's user avatar
1 vote
0 answers
43 views

Continuous function from a product topological space to a sum topological space

I denote by $\mathbf{2}$ the set $\{0, 1\}$ that is given the discrete topology. Let $X$ be some topological space. I denote by $\mathbf{2} \times X$ the topological space with the product topology ...
Bruno's user avatar
  • 288
0 votes
1 answer
40 views

Vector bundle associated to the universal cover $\mathbb{R}\to S^1$

It's a well known fact that, given a principal $G$-bundle (where $G$ is a Lie subgroup of $\text{GL}(r,\mathbb{R})$) $$\pi_P:P\to X$$ there is an associated vector bundle $$\pi_E:E(P):=(P\times \...
Kandinskij's user avatar
  • 3,666
1 vote
1 answer
34 views

Compactness of $[x-\epsilon,x+\epsilon]$ in topology consisting of $\tau_\text{Eucl} \cup \mathcal{P}(\mathbb{Q})$

I'm trying to show that $\mathbb{R}$ with the topology generated by $\tau_\text{Eucl} \cup \mathcal{P}(\mathbb{Q})$ is locally compact. To show that the topological space is locally compact in the ...
Bentley Edwards's user avatar
-1 votes
0 answers
54 views

Is there "meta" algebra, that operates with the collections of expressions?

Classical decision problem (https://web.eecs.umich.edu/~gurevich/Books/00.pdf - book, it opens as pdf) considers the sets of formulas which have decidable SAT and which have not decidable SAT problem. ...
TomR's user avatar
  • 1,321
-1 votes
1 answer
53 views

How to study three-dimensional manifolds through group presentation of three-dimensional manifolds? [closed]

I know the group presentation of a three-dimensional manifold, such as $$\pi_1(M)=\langle a,b\vert(ba)^2a^2,(ba)^2b^2\rangle.$$ Is there any good way to understand this manifold? Or which specific ...
ssjstchhh's user avatar

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