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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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3 views

Why the boundary set of a set can have more elements of the set in question?

We know by definition, the boundary point of $S \subset \mathbb{R}$ is any point $x$ (in every neighborhood of $N$) such that $N \cap S \neq \emptyset$ and $N \cap (\mathbb{R} \setminus S)$. This ...
0
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0answers
7 views

The proof of the homotopy extension and lifting property (HELP)

I am reading J. P. May's Book, the section about homotopy extension and lifting property (HELP) on page 75: I know this is true for $$(X,A)=(D^n,S^{n-1})\cong (CS^{n-1},S^{n-1}),$$ where $CS^{n-1}$ ...
1
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0answers
23 views

Any set of $\mathbb R$ is open iff it is the union of disjoint open intervals

I am starting in my studies on topology and I am having really big troubles with this kind of stuff. I hope it is easier than it looks
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1answer
10 views

Continuity of Projection operator from Tangent Bundle to Manifold

I've been struggling with this problem for several hours now and I would appreciate some assistance. This is the only part of the larger problem that I am stuck on. The Tangent Bundle $TM$ of a ...
0
votes
1answer
45 views

Having a hard proving $\{ t_i > 0 \mid \sum t_i = 1\}$ is open

Let $S = \{ (t_0, ..., t_n) \in \mathbb{R}^{n+1} \mid \forall i, t_i >0, \sum t_i = 1\}$. Is the set $S$ closed or open or neither ? I think the set $S$ is closed, yet it's hard for me to prove ...
0
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4answers
69 views

$M$ is metric space. $A\subset M, F$ is closed in $M$, prove $A\cap F$ is closed in $A$ [duplicate]

$M$ is metric space, $A\subset M, F$ is closed in $M$, I'm asked to prove $A\cap F$ is closed in $A$. First of all, what does "closed in $A$ mean"? Does it mean that $A$ is now the metric space ...
0
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0answers
25 views

Surjectivity of real continuous expansive-type functions

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ be continuous and let there exist $\alpha > 0$ such that $||f(\mathbf{x}) - f(\mathbf{y})|| \geq \alpha || \mathbf{x} - \mathbf{y}||$ for all $\mathbf{x}, \...
2
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1answer
51 views

Is $(S^1)^3/{\sim}$ a manifold? $\sim$ is equivalent relation on all triple permutations $(a,b,c)\sim (a,c,b)\sim (c,a,b)\sim … $ on $(S^1)^3$

Let $\sim_1$ be an equivalent relation on $(S^1)^2$, $(a,b)\sim(b,a)$, then $M_2:=(S^1)^2/{\sim_1}$ is homeomorphic to Mobius band and is a $2$-manifold,. What can we say about $M_3=(S^1)^3/{\sim_2}$ ...
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1answer
23 views

Continuous image of a locally path connceted space is locally path connected? [duplicate]

We already know that continuous image of path-connected spaces are path-connceted. I was thinking if the same result is true when we replace path - connected with locally path - connceted. I think I ...
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2answers
20 views

Interior of linear subspace

Suppose that $H$ is a Hilbert space and $K$ is a closed linear (strict) subspace of $H$. Then, is $$ \operatorname{int}(K)=\emptyset? $$ This seems to be the case, for example when I take $K\...
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1answer
25 views

Let $X=\Bbb{N} \setminus \lbrace 1 \rbrace$, $A_n=\lbrace d\in X : d|n \rbrace$, for $n\in \Bbb{N}$. Is $\tau=\lbrace A_n : n\in \Bbb{N}$ topology?

Let $X=\Bbb{N} \setminus \lbrace 1 \rbrace$, $A_n=\lbrace d\in X : d|n \rbrace$, for $n\in \Bbb{N}$. Is $\tau=\lbrace A_n : n\in \Bbb{N}\}$ a topology? My attempt: 1) It is obvious that $\emptyset, X ...
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2answers
45 views

Show that $\mathbb{R^{n}}$ is not compact by definition

I want to show by definition that $\mathbb{R^{n}}$ is not compact, so I did this: Let ${\Omega}$ be an open cover of $\mathbb{R^n}$ and suppose that $\mathbb{R^n}$ is compact, then there exists a ...
0
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2answers
29 views

Show that $\tau$ is strictly finer than the euclidean topology.

Let $A=\lbrace \frac{1}{n} : n \in \Bbb{N} \rbrace$, $H=\lbrace (a,b) \rbrace \cup \lbrace (a,b) - A \rbrace$. Let $\tau$ be the topology on $\Bbb{R}$ generated by the basis $H$. Show that $\tau$ ...
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0answers
37 views

Show that the topology on $X$ is contained in the Box topology on $X$

I have been searching and searching through the Munkres Topology book for examples and trying to understand how to do this problem but I have had no luck as of yet. If someone could help out it would ...
2
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1answer
29 views

How to show that $f(x)$ is not continuous using open sets?

I am practicing topology with a book, and I'm trying to understand the following definition. A map $f$: $\mathbb{R} \rightarrow \mathbb{R}$ is said to be continuous if for every open subset $U \...
0
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0answers
43 views

Proving that unit quaternions are a 3 Manifold

I am very new to topology, and I am having trouble on how to prove if something is a manifold or not. The question states that: Let Q donate the set of unit quaternions (a) Show that Q is a 3-...
0
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2answers
40 views

Is this proof that $\text{diam}(A) = \text{diam}(\bar{A})$ correct?

Let $M$ be a metric space. I'm asked to prove that the diameter of a set $A\subset M$ is the same diameter as its closure $\bar{A}\subset M,$ $$\text{diam}(A) = \text{diam}(\bar{A}).$$ My attempt: ...
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4answers
57 views

Is my proof that $\overline{A}\cup\overline{B} = \overline{A\cup B}$ correct?

Let $\overline{A}$ define the closure of $A$. I'm asked to prove that $$\overline{A}\cup\overline{B} = \overline{A \cup B}.$$ My attempt at this: $\overline{A}\cup\overline{B}$ is the union of the ...
1
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0answers
38 views

Some extension for connectedness

Let $X $ be a quasi compact topological space and $Y $ be a subspace of $X $. I am looking for some topological properties on $Y $ such that if $Y $ is a connected subspace of $X $, then $Y $ has ...
0
votes
2answers
40 views

A map that has no lift

I am trying to understand an example given in Chapter 8 of Lee's Introduction to Topological Manifold. If $B$ is a topological space and $\varphi:B\rightarrow\mathbb{S}^1$ is a continuous map, we ...
0
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2answers
23 views

An explanation as to why U equals the indiscrete topology on A.

Let $A\subset X$, and assume that $X$ has the indiscrete topology. If $U$ is the induced subspace topology on $A$, explain why $U$ is equal to the indiscrete topology on $A$. So far I understand that ...
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0answers
39 views

In TVS, is it true that every neighbourhood of zero is sum of some two other neighbourhoods of zero

Let $V$ be a neighbourhood of zero. Is it true that there always exists neighbourhoods of zero A and B such that $V=A+B$? If this is true, then could it be generalized to neighbourhood of any point, ...
0
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1answer
13 views

necessary condition for a $T_1$ space to be paracompact

If a $T_1$ space is paracompact then for every open cover $\{U_s:s\in S\}$ of $X$ there is a continuous mapping $f:X\rightarrow Y$ onto a metrizable space $Y$ and an open cover $\{W_s:s\in S\}$ of $Y$ ...
4
votes
2answers
87 views

Find finite path connected topological space with $π_1 (X, x_0 ) \cong \Bbb Z/2\Bbb Z$

Find a finite non-empty topological space $X$ that satisfies: $X$ is path connected Its fundamental group $π_1 (X, x_0 ) \cong \Bbb Z/2\Bbb Z$ for $x_0 \in X$ Is the connected pair $\{\{0\},\{1\},\{...
1
vote
1answer
33 views

Prove the equivalent conditions for nowhere dense subset.

Let $(X,d)$ be a metric space and $A$ be a subset of $X$. Then the following statements are equivalent. $A$ is nowhere dense. $\overline{A}$ doesn't contain any non-empty open set. Each ...
0
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1answer
32 views

Preimage of continuous one-to-one function on connected domain is not continuous.

I know that given $B$, a compact subset of $\mathbb{R}^n$, and $f : B \to \mathbb{R}^m$, a continuous injective (one-to-one) function, $f^{-1}$ is continuous on $f(B)$. (This true). I also know that ...
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0answers
34 views

I am having trouble proving this [on hold]

Prove that if $x \neq (0,0)$, then the singleton $\{x\}$ is open
1
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2answers
52 views

Proving that a particular set in $\mathbb R^2$ is open.

I have to prove that the subset $A=\{ (x,y)|x^2>y\}$ of $\mathbb R^2$ is open. I graphed it, and I can see that it is open, and I can also see that the boundary is $y=x^2$. We have not yet ...
0
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3answers
20 views

Show that $\operatorname{Cl}(A)\setminus A$ consists of entirely of accumulation points of $A$

Let $(M,d)$ be a metric space and $A \subset M$. Show that $\operatorname{Cl}(A)\setminus A$ consists of entirely of accumulation points of $A$, where $\operatorname{Cl}(A)$ is the closure of $A$. My ...
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0answers
22 views

How to calculate the algebra of chord diagram with 4 chords? [on hold]

can somebody please explain How to calculate the algebra of chord diagram with 4 chords? many thanks!
0
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0answers
24 views

How to prove that the mapping of the braid to the Automorphism group of the free group is a complete invariant?

Does anybody know how to prove that the mapping of the braid to the free group's automorphism group is a complete invariant?
-1
votes
0answers
40 views

Product of continuous functions [on hold]

Let $(\mathcal{U}_{n}:n\in \mathbb{N})$ be a sequence of countable covers of $X$ by co-zero sets. For every $n\in \mathbb{N}$ and every $U\in \mathcal{U}_{n},$ fix a continuous function $f_{U}:X\...
0
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1answer
18 views

Prove that two variable Jones polynomial can be expressed by Finite type invariant

I have this question that says: Prove that two variable Jones polynomial can be expressed by Finite type invariant. Can somebody explain how this is done? many thanks in advance!
0
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0answers
15 views

Show that $K∩L$ is homeomorphic to $A$, where $K ∩ L = \{\{(a,0),(a,1)\} : a \in A\}$ is a subset of equivalence classes in $[0,1] \times \{0,1\}$.

I'm trying to solve the following question: Let $X$ be a topological space and $A$ a subset of $X$. On $X\times\{0,1\}$ define the partition composed of the pairs $\{(a,0),(a,1)\}$ for $a\in A$, ...
0
votes
6answers
51 views

$x \neq y\in$ metric space $M$, prove $\exists$ open sets $U,V$ s.t. $x\in U,\ y\in V$ and $\bar{U} \cap \bar{V} = \emptyset$

Let $x,y\in M, \ x\neq y,\ M \ \text{being a metric space}$. I'm asked to prove that there exists open sets $U,V\subset M$ such that $x\in U,\ y\in V \ \text{and} \ \bar{U}\cap\bar{V} = \emptyset$...
0
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0answers
29 views

Help proving this map is closed

Define the following equivalence relation on $\mathbb{R}^2$: $(x,y)\sim(x',y')$ iff there is $n\in \mathbb{Z}:(x',y')=(x+n,(-1)^ny)$. Let be $E=\mathbb{R}^2/\sim$ the quotient space, and $q:\mathbb{R}^...
0
votes
2answers
67 views

What can we say about two topological spaces with the same fundamental group?

Let's consider two topological spaces. If they are homeomorphic, or homotopic equivalent, they have isomorphic fundamental groups, but the converse is not true. My question is: is there a (non trivial)...
0
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1answer
43 views

Punctured open set is not contractible

Let $U\subseteq\mathbb{R}^2$ be an open subset, and let $x\in U$. Then $U\setminus\{x\}$ is not contractible. A space $X$ is called contractible if the identity map on $X$ is homotopic to a constant ...
5
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0answers
50 views

If $\gamma\colon[a,b]\to\mathbb{C}$ is continuous and $\gamma(b)=-\gamma(a)$, must the curves $\gamma$ and $e^{ic}\gamma$ intersect for all real $c$?

If $[a, b]$ is a compact interval of $\mathbb{R}$ and $\gamma: [a, b] \to \mathbb{C}$ is continuous, denote the connected, compact set $\gamma([a, b])$ by $[\gamma]$. If $h$ is a complex number of ...
0
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1answer
15 views

Dictionary order topology based query

I am not sure i fully understand dictionary order topology...first picture still confusing me, why there is line in upward of a*b , why its jot downward. and what will be diagram for (2*2,4*5).
2
votes
3answers
50 views

Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?

Let $f:X\to X $ be continuous. Show that if $X=[0,1] $, $f $ has a fixed point(i.e. there exists $x$ such that $f (x)=x$). What happens if $X $ equals $[0,1) $ or $(0,1) $? First part of the ...
3
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0answers
53 views
+50

Surjective functions from a $n$-dimensional hypercube to $\mathbb{R}^m$ when $n > m$

I had asked a similar question before. Functions from an $n$-dimensional hypercube to $\mathbb{R}^m$ when $n >m$. I am wondering if there are any surjective functions. Let $n$ and $m$ be ...
0
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1answer
43 views

Surjections between homology groups

When does there exist a continuous surjection between two cell complexes $X$ and $Y$ such that $H_*(X)$ is not isomorphic to $H_*(Y)$. What properties must be satisfied?
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2answers
50 views

Is this a typo about the derived set in textbook Introduction to Set Theory by Hrbacek and Jech?

In my textbook, $A'$ is the set of all limit points of $A \subseteq \Bbb R$. I think the below statement is possibly not correct. For each $\alpha$, the set $F^{(\alpha)} - F^{(\alpha+1)}...
0
votes
2answers
34 views

How does the definition of an open ball in a metric space exclude the empty set?

Definition. Let $(X,d)$ be a metric space. An open ball of radius $r>0$ around a point $x\in X$ is the set $B(x,r) = \{ y \in X \, | \, d(x,y)<r \}$. Looking at the definition of an open ball, ...
3
votes
1answer
53 views

T shaped tetris figures on a plane

I am just wondering how many (and by how many I mean countably or uncountably many) T shaped figures can we place on a XY plane. I assume that that T consists of 2 perpendicular lines and has 0 area. ...
0
votes
1answer
27 views

Preimage of $\partial\Omega$ under a continuous function

Suppose $\Omega\subset \mathbb{R}^n$ is open and bounded, and let $\partial\Omega$ the boundary of $\Omega$. Fix $x \in \overline{\Omega}=\Omega \cup \partial\Omega$ and let $y:[0,+\infty)\to \mathbb{...
0
votes
1answer
42 views

Square Torus homeomorphism

If we have a square torus on R$^2$ defined by the equivalence relation $(a_1, b_1)$ ~ $(a_2, b_2)$ if and only if $a_1 - a_2$ and $b_1 - b_2$ are integers, how would you show that they are ...
1
vote
1answer
19 views

(Covering space) Does Deck move whole sheets in whole sheets?

Let $p: E \to B$ be a covering space. Let be $b \in B$, $U \in I(b)$ an evenly covered neighborhood of $b$, $U_i$ the sheets over $U$ such that $p^{-1}(U) = \coprod{U_i}$. If $\phi \in Deck(p)$, is ...
0
votes
0answers
43 views

How to prove that the action of two dimensional torus is Hamiltonian?

Given the 2 dimensional torus: $T^{2}={(\varphi_{1} \mod 2\pi, \varphi_{2} \mod 2\pi)} $ that acts on $CP^{1}\times CP^{1}$ by: $ \left(\varphi_{1},\varphi_{2}):((z_{0}:z_{1}),(w_{0}:w)\right)) \to ...