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; ...

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18 views

Let $X=\mathbb R$ with the usual topology. If $A=[0,1]$, how can $0$ or $1$ be an interior point of $A$?

Let $X=\mathbb R$ with the usual topology. Every subset of $X$ is clopen, so for every $Y \subset X$, int($Y$) $= Y$. But if $A=[0,1]$, the neighborhood of $0 \in A$ or $1 \in A$ is not a subset of ...
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7 views

Prob. $ 9 $, Sec. $ 23 $ of Munkres’ “Topology”, $ 2^{\text{nd}} $ Ed.: How to show this subspace is connected?

Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, how to show that $$\left(X \times Y \right) \setminus \left(A \times B \right)$$ is also ...
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1answer
20 views

The set of points of continuity of a real-valued function on a metric space is a $G_\delta$ set

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. I know lots of other ...
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2answers
27 views

The distance between two sets does not change if closure is taken

Given $ (X, d)$ a metric space, $ A, B \subset X$, show that $ d(A, B)=d (\overline {A}, B) $. I'm not being able to show that $ d(\overline{A},B) \leq d (A, B) $. Can anybody help me? The set ...
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16 views

Basis for a topology of a scheme

Suppose that $X$ is a proper and connected scheme over an algebraically closed field. Moreover let $\mathcal A$ be a collection of open subsets of $X$ with the following property: For every open ...
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1answer
38 views

Some question about path connectedness

I think that's intuitively evident but I can't prove that the set $\mathbb{S}^n\setminus\{(0,\cdots, 1),(0,\cdots, -1)\}\; (n>1)$ is path connected. Does anyone have a formal argument to prove it? ...
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17 views

Regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $.

I tried to draw the regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $. I think the regular covering space is: Is it true? How do you draw the non-regular ...
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11 views

Manifold topology vs. trace topology induced by the ambient space

I'm confused about the topology of submanifolds of $\mathbb{R}^n$: Let $M$ be such a $k$-manifold (say, the circle $S^1$, of dimension $1$, embedded in say $\mathbb{R}^7$); the topology of such a ...
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1answer
20 views

Prove that a function is open

Let $X,Y$ metric spaces and $U \subset X , V\subset Y$ open sets. Let $f:U\rightarrow V$ be a homeomorphism. Prove that $f$ is an open map. I need to show that for every open subset of $U′⊂U$, $f(U′)$ ...
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1answer
35 views

Let $A = \{1- \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$.

Let $A = \{1 - \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$. I am supposed to figure out if this set is closed under certain topologies. I know that means I ...
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2answers
13 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
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0answers
14 views

Operator norm on the space on linear functions between Euclidean spaces.

*I'm reading a text which has a preliminary section on Linear maps. I have come across a conclusion that I can't seem prove by myself. * Let $Lin(\mathbb{R}^m,\mathbb{R}^n)$ be the space of linear ...
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3answers
194 views

Confusion about concept of basis in point set topology.

I'm afraid that I have a big misunderstanding about the notion of basis in general topology. For a given topology $\tau$ of set $X$, if there is a collection $S \subset \tau$ of open subsets of $X$ ...
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1answer
73 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
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31 views

Probs. 2 (d) and 2(e) in Supplementary Exercises, Chap. 2 in Munkres' TOPOLOGY, 2nd ed: How are these maps continuous?

Let $S^1$ denote the set of all complex numbers $z$ such that $\vert z \vert = 1$ (regarded as a subspace of the complex plane), and let the map $f \colon S^1 \times S^1 \to S^1$ be defined by $$f(w ...
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2answers
145 views

Do two closed subsets of $[0, 1]$ with measure $\frac{1}{2}$ intersect?

Let $A$ and $B$ be two closed subsets of $[0,1]$, each with a length of $1/2$. Is it always true that $A\cap B\neq \emptyset$? My intuition is yes, because: Either they intersect in their interior; ...
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1answer
17 views

Continuity of product maps

Let $J$ be a given (countably or uncountably infinite) index set. Let $\{\ X_\alpha \ \colon \ \alpha \in J \ \}$ and $\{\ Y_\alpha \ \colon \ \alpha \in J \ \}$ be collections of topological ...
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22 views

Show if orbit is discrete, the orbit is closed.

Given $\beta <$ the isometry group $\mathbb{R}^2$. Show that if an orbit $\beta_x$ is discrete, then $\beta_x$ is closed. I am just looking for some feedback and critique of my attempt at a ...
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1answer
22 views

Is there any simply connected polyhedron with a not simply connected face?

According to Wikipedia, For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2. Is it really necessary to specify here, that ...
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1answer
45 views

Comparing different topologies on the Hilbert cube $H = \prod_{n \in \mathbb{N}} [0,\frac 1n]$

This is essentially exercise 8(c) from section 20 (p.128) of Munkres's Topology: Let $X$ be the set of all the sequences $(x_n)$ of real numbers such that the series $\sum_{n=1}^\infty x_n^2$ ...
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2answers
32 views

Is every point in the set defining some curve or line in $\mathbb{C}$ or $\mathbb{R}$ a boundary point? [on hold]

Might be a dumb question but is every point in the set defining some curve or line in $\mathbb{C}$ or $\mathbb{R}$ a boundary point? I reason that it should be since any point in the set has to be ...
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2answers
31 views

Construct an open cover of S with no finite subcover

Let S be a subset of Rn, and suppose that S is not bounded. Construct an open cover of S with no finite subcover, then prove this claim about your open cover. Let S be a subset of Rn such that S is ...
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1answer
31 views

Prove that $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$ is compact and connected

Let be $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$. I have proved that $X$ is a submanifold of $\mathbb{R}^4$ of dimension $3$. I have to prove that $X$ is compact and connected. My idea, thinking of ...
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3answers
46 views

how do you prove the set of accumulation points of Q is R.

I know that the set of accumulation points for the rational numbers is the real numbers, but I'm not sure how to prove this. I need to use the definition: $x$ is an accumulation point of $S$ if, for ...
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1answer
18 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
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23 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
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1answer
13 views

Let $(X, \mathfrak T)$ be topological space and suppose that A and B are subsets of X such that $A \subsetneq B$. Then $Int(A) \subsetneq Int(B)$.

Let $(X, \mathfrak T)$ be topological space and suppose that A and B are subsets of X such that $A \subsetneq B$. Then $Int(A) \subsetneq Int(B)$. ( $\subsetneq$ means "is a proper subset") My ...
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1answer
34 views

Fractal dimension of a dense subset

Let $M$ be a metric space and $S\subset M$ a dense subset. For vague reasons (below), it seems to me that the upper box-counting dimension of $S$ should be equal to that of $M$, but I don't quite see ...
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26 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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1answer
45 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then A =∅ or $A = X$ .

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then $A = \emptyset$ or $A = X$. I am studying introduction to proofs and we have learned ...
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0answers
38 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...
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1answer
23 views

K is convex.Is this true that ${K^ \circ } \ne \phi$?. [on hold]

Let $ K$ is convex.Is this true that ${K^ \circ } \ne \phi$?
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131 views

A new proof of Tychonoff's theorem from the subbase theorem for total boundedness

The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) : Theorem: Let $(X,\mathcal{U})$ be a ...
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1answer
51 views

Complete metric space of sequence of positive integers [duplicate]

Let $(A,d)$ be the space $\mathbb{N}^{\mathbb{N}}$ of sequences of positive integers where $d((a_i)_i, (b_i)_i)= \frac{1}{n}$ where $n$ is the least coordinate at which $(x_i)_i$ and $(y_i)_i$ ...
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1answer
22 views

Interiors and Closure

I want to prove $int(\overline S)\subset \overline {S \cap int(\overline S)}$ I would like some hints(Only hints I dont want the solution)
2
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1answer
45 views

Background required to understand the mathematical definition of knots and their transformations

What are the concepts of math required as a prerequisite to understand Knot Theory? I'd like to be able to make a humble beginning by being able to mathematically define knots and the non-rigid ...
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1answer
35 views

Homeomorphism between normal space and subspace of $[0,1]^k$

Given $A$ is normal and has a basis $B$ in which $|B|=k$. Show that $A$ is homeomorphic to a subspace of $[0,1]^k$. My Thought I was thinking of applying the Jones's lemma here, but I couldn't even ...
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50 views

How to think/see point-set topology abstractly?

I've started learning point-set topology this semester. I've learned basic material about: topology on a set topological space open sets closed sets clopen sets closure neighborhoods interior point ...
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13 views

Topological Entropy and generator: Do we need that T is a homeomorphism?

There is the following statement in Walters concerning the computation of Topological Entropy in case of an expansive homeomorphism: Let $T\colon X\to X$ be an expansive homeomorphism on a compact ...
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2answers
83 views

Visualizing products of $CW$ complexes

I'm learning about products of CW complexes. The sources I've seen talk about the matter as follows: given topological spaces $X$ and $Y$ with a given CW decomposition, we can then form a CW ...
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1answer
33 views

Infinite set on finite closed topology

I'm having trouble making sense of certain terminology. So the question asks me to determine whether a finite closed topology on an infinite set is a $T_1$ space. Now before I get into that, I'm ...
6
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32 views

Every net has an ultranet as subnet: direct proof

I'm currently brushing up my topology using Willard's General Topology. Currently I'm working through the chapters 11 and 12 on nets and filters. Chapter 12 deals extensively with ultrafilters and ...
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1answer
76 views

A $ 0 $-dimensional topological subspace of $ \Bbb{R}^{\Bbb{N}} $.

Given the space $X(\mathbb{R}^{\mathbb{N}}, i)$ contains points $(x_{j})_{j\in N}$ of $\mathbb{R}^{\mathbb{N}}$ with only $i$ coordinates rational. (a) Show that $X(\mathbb{R}^{\mathbb{N}}, 0)$ is ...
3
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1answer
39 views

Metrizability of a topological space

If we have a topological space $(X,\mathcal{T})$ and a metric $d$ on $X$ s.t. for any sequence $(x_n)_n$ convergence of the sequence $x_n$ to some $x \in X$ for the topology $\mathcal{T}$ is ...
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1answer
44 views

Existence of open set in a continuous mapping

Give $f: A\rightarrow B$ is a closed , onto and continuous mapping. For $b\in B$, let $C$ be an open set in $A$ s.t $\ f^{-1}(b)\subset C$. Show that there exists open set $D$ in $B$ with $b\in D$ s.t ...
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1answer
22 views

Showing topological properties of a function

Let $f$ be a function from a set $X$ into a set $Y$. prove: i) the function $f$ has an inverse if and only if $f$ is bijective ii) let $g_1$ and $g_2$ be functions from Y into X. If $g_1$ and $g_2$ ...
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1answer
13 views

What is the difference between a period $n$ point, and a point of least period $n$?

What is the difference between a period $n$ point, and a point of least period $n$? Simply what is the definition of the two of them, and how do they differ. I think I have a rough idea of one ...
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2answers
29 views

Are all retracts with the same homotopy type as the bigger set, deformation retracts?

I was reading the book of james r munkres about topology and noted that if A is a deformation retract of X, A and X need to have the same fundamental group. But is a retract that has the same ...
0
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1answer
21 views

A property of a bounded set

Prove that is $A$ is bounded, then the least upper bound of $A$ is in $A$ or a point of accumulation. proof: Let $A$ be a bounded set and $x\in A$. Then, since $A$ is bounded there exists an ...
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1answer
43 views

Metrizability of the countable product of metric spaces

We know (Show that the countable product of metric spaces is metrizable) that the following is true: Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$. Form the ...