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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2
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38 views

Topological features (and / or definition) of homology

I am coming to grips with basic homological algebra as of late, in order to better understand my own subject, that of the study of language. The thing is, I have recently read in some handbook that ...
1
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2answers
37 views

Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
3
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2answers
56 views

Fundamental group of $\mathbb{R}^n\backslash \{0\}$

I am wondering about what the fundamental group of $\mathbb{R}^n \backslash \{0\}$ or more generally $\mathbb{R}^n \backslash U$ where $U$ is a subset of $\mathbb{R}^n$ for $n>1$. For $n=1$ I ...
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0answers
29 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
2
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0answers
35 views

Question about contractible set .

Please if i have a contractible and closed set $A$ in $X$ thene $A$ is closed and there existe a continuous function $H:[0,1]\times A\rightarrow X$ such that $H(0,u)=u, H(1,u)=p\in X.$ If i ...
3
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0answers
37 views

Defining a topological relationship between two objects

I am looking for a mathematical definition/description of the following relationship between two objects. It's similar to a knot (as in topology) but between two objects. I've found a similar problem ...
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1answer
71 views

If X is limit point compact space,which is T1,then X is countably compact.

Countably compact means : every countable open covering contains a finite subcollection that covers it. Limit point compact means: every infinite set contained in it has a limit point. In T1 space ...
5
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2answers
180 views

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
1
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1answer
31 views

Set of all real numbers with the Scott topology

It is known that a space $X$ is compact iff every net in $X$ has a cluster point. Let $\sum\mathbb{R}$ be set of all real numbers with the Scott topology. I know that $\sum\mathbb{R}$ is not compact. ...
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1answer
38 views

How to show that this set is closed in $\mathbb{R}^n$?

For an open set $\Omega\subseteq\mathbb{R}^n$, let $K_j$ be the set of points $x$ of $\Omega$ such that $\text{dist}(x,\partial\Omega)\geq1/j$ and $|x|\leq j$. Question : Why is $K_j$ closed ? ...
2
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1answer
57 views

$\mathfrak{Top}$ and injective objects

My question is very simple. Let $\mathfrak{Top}$ be the category of all topological spaces and continuous functions between them. Does such category have enough injectives? Is there a simple way to ...
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2answers
50 views

Induced subgroup of $\pi_1(S^1)$ by $p_n$

Consider the following covering map $p_n: S^1 \to S^1, z \mapsto z^n$. Why is the subgroup of $\pi_1(S^1)$ induced by $p_n$ isomorphic to $n\mathbb{Z}$? I know that $\pi_1(S^1) \cong \mathbb{Z}$ but ...
0
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1answer
34 views

There is no metric d,so that (Q,d) is a connected space [duplicate]

Can anyone prove this? There is no metric d,so that (Q,d) is a connected space Q are rationals.
4
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1answer
70 views

Locally Compact Spaces: Characterizations

For Hausdorff spaces the following are equivalent: Every point admits a compact local base. Every point admits a compact neighborhood. Every point admits a precompact neighborhood. Every point ...
4
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3answers
114 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
3
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2answers
98 views

Construct a set of real numbers whose limit points comprise the set of integers $\mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only ...
0
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0answers
73 views

Cantor set--nowhere dense, complete

I can't figure out this out. Cantor set is closed in $\mathbb{R}$. $\mathbb{R}$ is a complete metric space. Every closed subset of a complete space is also complete; thus, so is the Cantor set. ...
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0answers
53 views

Separable spaces and functions that separate points

In a metric space, does existence of a function that separates points imply that the space is separable and conversely? I'm just a baby Rudin student. Thanks in advance for every hint.
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1answer
65 views

Can interior set or exterior set be empty?

I'm trying to prove or disprove that if $X$ is a boundary of $S$ in $R$, then every ball $B(x)$ contains both interior point of $S$ and exterior point of $S$. I'm trying to think of counterexample, ...
1
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1answer
169 views

If $E$ is a closed set there exist a set $S$ such as $E=S'$

In "Elementary Real Analysis" by Thomson-Bruckner p.190 I did the following exercise: (we're working on $\mathbb{R}$ and elementary topology on that set) One of Cantor's early results in set ...
1
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1answer
63 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
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1answer
41 views

How does one see connectedness of a covering space?

Something can be proven about the loops (or their possible lifts?) in the base space which will ensure connectivity of the cover?
2
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2answers
50 views

Given $S \subset \Bbb{R}$, show $\textbf{int}(S)+\textbf{ext}(S)+\partial S =\Bbb{R}$

The way I proved it is that we knwo R is open so intR=R. For any point in IntS is inside of IntR and any point in ExtS is inside of IntR. any point that is neither intS nor extS is still inside of ...
2
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0answers
46 views

In what way is the representation not continuous

http://www.math.u-psud.fr/~fontaine/galoisrep.pdf pp.7-8 Following the definition of a linear representation it states 'if V (an E-vector space) is equipped with a topological structure and if ...
0
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1answer
32 views

augmented chain complex

From Hatcher's Algebraic Topology, I know that a continuous map induces a morphism of chain complexes $f :C(X) → C(Y)$ by invariance of homotopy, but how would I show that $f$ also induces a ...
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2answers
57 views

The set of all exterior points is an open set

Let $S$ be a subset of $R$. Then the set of all exterior points of $S$ is an open set. My proof is as follows: For any element $x$ in $\operatorname{ext}(S)$ (the set of all exterior points ...
3
votes
2answers
140 views

Topology on $Z_p$

let $Z_p$ denote the $p$-adic integers, then it has a topology as a subspace of $\prod_nZ/p^nZ$, where $Z/p^nZ$ is given the discrete topology. (reference I posted before: Why $Z_p$ is closed.) Now ...
0
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2answers
88 views

Is my proof that empty set is open and R is open correct?

Claim: The empty set is open. Proof. Assume that the empty set is closed. Then, there must be one point such that any point in its ball is not inside of the empty set. However, the empty set has no ...
2
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1answer
55 views

Acyclic model type result: existence of a chain homotopy

If $\sigma: \Delta_n \to X$, define $\overline{\sigma}: \Delta_n \to X$ by$$\overline{\sigma}(t_0, \dots, t_n) := \sigma(t_n, \dots, t_0).$$Define a map $T: C_n(X) \to C_n(X)$ by $T(\sigma) := ...
2
votes
1answer
79 views

$\mathbb{A}^n$ with the Zariski Topology is Quasi-Compact.

I want to show that $\mathbb{A}^n$ is quasi-compact. I'm kind of stuck, I really don't know where to go with my proof, so I'll show what I have Proof: So suppose that $\cup U_i$ was an open cover for ...
2
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3answers
58 views

Show that the Euclidean Metric is less than or equal to the Taxi-cab metric for $\mathbb{R}^{n}$

I am trying to prove that the Euclidean metric; $(\mathbb{R}^{n},d^{2})$; defined: $$d^2(x,y) = \sqrt{\sum_{i=1}^n(x_i-y _i)^2}. $$ is less than or equal to the Taxi Cab metric; ...
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3answers
77 views

Exercise on Metric space

I hve this exercise it is very simple but i don't know how to write the answer Let $A$ be a nonempty set in $(E,d)$, for $\varepsilon>0$ we note $$V_{\varepsilon}(A)=\{x\in E, ...
0
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0answers
45 views

How to convert an object into a sphere?

I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a ...
0
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2answers
46 views

Openness of inverse limit of quotient mappings

Let $X$ be a topological space. Let $(R_\alpha)_{\alpha \in A}$ be a family of equivalence relations on X, whose index set $A$ is directed. We have canonical mappings $\phi_\alpha : X \to X/R_\alpha$. ...
0
votes
1answer
47 views

Requirements for Mayer-Vietoris

This question might be a duplicate -- but as I don't find an entry (maybe because of the lack of a good keyword) I open this question. Besides, this questions arises when trying to prove Proposition ...
0
votes
1answer
39 views

Check $S\cap T$ where $S =\left\{ x \in \mathbb{R} : x^6 -x^5 \le 100\right\}$ and $T =\left\{x^2-2x : x \in (0,\infty)\right\}$

Let $S=\left\{x\in\mathbb{R} : x^6 -x^5 \le 100\right\}$ and $T=\left\{x^2-2x : x \in (0,\infty)\right\}$. Then check whether or not $S\cap T$ is Closed and bounded in $\mathbb{R}$ Closed but not ...
0
votes
1answer
29 views

Topology generated by a Family of Seminorms as a Initial Topology?

Let $X$ be a set and $\{(Y_i, \mathscr{T}_i)\}_{i\in I}$ be a family of topological spaces and $\{f_i\}_{i\in I}$ a family of mappings $$f_i:X\longrightarrow Y_i.$$ The initial topology on $X$ is the ...
0
votes
1answer
48 views

Separability of the Set of Bounded Functions over [0,1]

I'm working through Neal Carothers' Real Analysis and I'm stuck on trying to show that the set $B$ of bounded, real-valued functions over $[0,1]$ is not separable. The metric of this set is ...
2
votes
1answer
24 views

Question About Assumption Relating to Bolzano-Weierstrass

We had the first day of our topology course today, and the instructor presented the Bolzano-Weierstrass Theorem on $\mathbb{R}$. I took it as a challenge to see if I could do it while he was talking, ...
2
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2answers
64 views

Show that $\mathbb{A}^n$ on the Zariski Topology is not Hausdorff, but it is $T_1$

There was an exercise I could not do. So the property is $T_1$ if for every pair of distinct points, $P, Q \in X$, there is an open subset $U$ containing $P$ but not $Q$ and another open subset $V$ ...
2
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2answers
75 views

Why $Z_p$ is closed.

Let $A_n=\mathbb{Z}/p^n\mathbb{Z}$ be a ring and $p$ is prime, $\phi_n: A_n\rightarrow A_{n-1}$ be a natural homomorphism (Elements of $A_{n}$ define in an obvious way elements of $A_{n-1}$). Define ...
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0answers
42 views

Any limit definition defines topology

Let $X$ be a normed space and let $C(X)$ be the space of the continuous functions $X\to R$. We have the dual space $C\left(X\right)^{*}$. Suppose we have the definition of limit on the dual space. ...
2
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2answers
129 views

The space $x^3-y^2=0$

Consider $\{(x,y)\in\mathbf{R}^2 \ | \ x^3-y^2=0\}$ as a subspace of $\mathbf{R}^2$. Intuitvely I understand that this is not supposed to be a differentiable manifold because it has a cusp at $0$. But ...
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1answer
38 views

How do I give a homeomorphism between $\mathbb R P^n$ and the space obtained by identifying antipodal points of $S^{n+1}$?

Suppose that $Y$ is the quotient space obtained by identifying the antipodal points of $S^ {n+1}$. I'm trying to give a homeomorphism between $\mathbb R P^n$ and $Y$. I think that the map ...
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0answers
24 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
5
votes
1answer
135 views

Why does Seifert-Van Kampen not hold with $n$-th homotopy groups?

My question concerns the Seifert-Van Kampen theorem, in the following form. Let $X$ be an arch-wise connected topological space, consider a poin $x_{0}\in X$, and let $\{U_{i}\}_{i\in I}$ be an open ...
3
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0answers
177 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
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1answer
26 views

What's the meaning of the state space with locally compact topological space?

I have encountered a statement in one paper describing the continuous-time controlled Markov chain with space state which is locally compact topological space. What does this mean? In my previous ...
0
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1answer
43 views

Accumulation point proofs

I understand accumulation points are basically limits and how to look at an accumulation point when given a numerical sequence but I'm having trouble finding some way of proving the following two ...
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2answers
22 views

When is the quotient space of a second countable space second countable?

I am a bit confused about this concept because I have read that the quotient space is second countable if the quotient map is open. However, I thought the definition of a quotient map was a ...