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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; ...

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Does there exist a continuous 2-to-1 function from the sphere to itself?

I am interested in the following question: Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$? I suspect the answer is no, but I don't know ...
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14 views

Set of Matrices with a given characteristic polynomial is compact [duplicate]

Consider the set of $3\times3$ matrices with the characteristic polynomial $x^3-3x^2+2x-1$. Is the set compact in $M_3(\mathbb{R})\cong \mathbb{R}^9$? The given characteristic polynomial has probably ...
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1answer
30 views

Whether the set $A$ is a compact subset of $M_3(\Bbb R)$.

Consider $$X=\Big\{A \in M_3(\Bbb R): \rho_A(x)=x^3-3x^2+2x-1\Big\}$$ where $\rho_A(x)$ is the characteristic polynomial of $A$ and $M_3(\Bbb R)$ is the space of all $3 \times 3$ matrices over $\Bbb R$...
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1answer
12 views

Local Path Connectedness - Collection Of All Path Connected Open Sets Is A Basis

A space $X$ is locally path connected if $X$ has a basis of path connected open sets. A follow-up to this question: Hatcher Universal Covering Space Construction - Basis From this definition, $X$ ...
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1answer
26 views

Any countable set has measurable zero

To demonstrate that ANY countable set has measure zero, is it sufficient to show that the natural numbers have a measure zero? If so, why; and, if not, why not? Thank you :)
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3answers
42 views

When is a sum of continous functions continous?

Let $(f_{\alpha})_{\alpha \in A}$ be a family of real valued continous functions on a topological space $X$ such that for each $x\in X$ $f_\alpha (x)\neq 0$ for only finitely many $\alpha\in A$. Then ...
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2answers
22 views

Separation Proof

I have been banging my head against this proof for a few days now, as I can visualize why it is true in my head, but don't know how to prove it in words: Let $A$ and $B$ be nonempty subsets of $\...
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2answers
31 views

Hatcher Universal Covering Space Construction - Basis

Below is an excerpt from Hatcher's Algebraic Topology. He is constructing a universal cover for a path-connected, locally path-connected, and semilocally simply-connected space $X$: I don't ...
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1answer
32 views

Homeomorphism between [0,1]/~ and the Hawaiian Earring

Let $X$ be the quotient space [0,1]/~ where 0 ~ 1 ~ 1/2 ~ $\cdots$ ~ 1/n ~ $\cdots$ Let $H$ (the Hawaiian Earring) be the subspace of $\mathbb{R}^2$ consisting of the union of circles of radius 1/n ...
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1answer
35 views

Properties of Cantor set

$[0,1]$ is not homeomorphic to $[0,1]×[0,1]$ but $C$ is homeomorphic to $C \times C$ where $C$ is the Cantor set. I know both the proof. I am asking which property of $C$ is the reason of this ...
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1answer
39 views

Uniform Continuity in Topological Spaces?

Is it already in literature this generalized notion of uniform continuity in an arbitrary topological space (not necessarily in exactly the same form)? Let $(X, T_{1})$, $(Y, T_{2})$ be topological ...
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Generalization of topologies with equivalence classes of sets

Is there a generalization of topological spaces which works on equivalence classes of subsets? To be a little bit more precise, I would think of something like the following: Let $X$ be a set and $P(...
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1answer
30 views

cluster and convergence points

I want to prove that if $x$ is a point of convergence, then it is also a cluster point. By exhibiting the proof, I want to understand where the proof gets stuck the other way around, i.e., that a ...
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1answer
24 views

Topology of decomposition of a space

In pg 121 of this notes, the author outlines a construction of gluing bundles. The scenario begins with Let $X= X_0 \cup X_1$ be union of two comapct spaces. $A = X_0 \cap X_1$ so that $X = X_0 \...
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1answer
30 views

Frontier properties of a subset of $\mathbb{R}$

I am supposed to prove or disprove the following claims: If $A \subset \mathbb{R}$ arbitrary, then $(Fr(Fr(A))^{\mathbb{o}} = \emptyset$ If $A, B \subset \mathbb{R}$ arbitrary, then $Fr(A \times B) ...
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3answers
45 views

find a free ultrafilter on $\Bbb N$

Suppose $(x_n)_n$ is a bounded sequence of complex numbers, there must exist a accumulation point, say $x_0$, thus we can find a free ultrafilter $\mathcal{F}$ on $\Bbb N$ such that $\lim_{\mathcal{F}}...
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1answer
50 views

The smallest compactification for continuous extension of $\sin(x)$

I would like to ask what the smallest Hausdorff compactification of $\mathbb{R}$ is, onto which we may continuously extend the function $g: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto \text{sin}(x)$. ...
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1answer
30 views

Is this argument about sequential compactness wrong?

In the 4th line of the first example of this page (it's a simple self-contained argument, I just doubt its validity), http://www.math.unl.edu/~s-bbockel1/929/node16.html they seem to be arguing that $...
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1answer
26 views

Hatcher Lemma 1A.3 - Covering Spaces Of Graphs Are Graphs

Below is an excerpt from Hatcher's Algebraic Topology: There are a few things I don't understand about the highlighted red part. What exactly is a "basic open set"? The highlighted sentence seems to ...
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1answer
32 views

A sequence of rationals converging to an irrational point (proving that $\mathbb Q$ is not locally compact)

Here is my attempt to prove that $\mathbb Q$ is not locally compact. (My questions are below the proof.) Suppose $\mathbb Q$ is locally compact. Then it is locally compact at every point. Let $x\in \...
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27 views

Computing homology groups

Given two copies of $D^2 \times S^1$ (Full torus) glued along the boundaries by a map from a Torus to itself defined by the action of $M \in SL(2, {ℤ})$; i.e., the map is $(x,y) \mapsto (x^ay^b, ...
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26 views

acculation point [on hold]

If $(x_n)$ is a complex sequence and we know that $a$ is point of $(x_n)$.By the definition of accumulation point in topological spaces ,for any neighborhood $U_a$ of $a$, $(x_n)\cap( U_a\setminus\{a\}...
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(Co)homology of $S^2×S^2/ℤ_2$

Cohomology of $S^2\times S^2/\mathbb{Z}_2$ I was looking at this question, the accepted answer uses the homology of the space to find the cohomology. I was wondering how one could compute the ...
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1answer
55 views

Proof of Arzela's Theorem

I am doing problem 3 from section 45 in Munkres. The problem is Prove Arzela's Theorem, which states: Let $X$ be compact: let $f_n \in \mathcal{C}(X,\mathbb{R}^k)$. If the collection $\{f_n\}$ is ...
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0answers
20 views

How to compute the local degree for a specific function

Using the definition of the local degree from Hatcher pg. 136, how can we explicitly calculate the local degree of a map? For example, to calculate the local degree of a map $f:R^2 \rightarrow R^2$ ...
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42 views

If $A\subset B$ and $B=V_1\cup V_2$, where $V_1$ and $V_2$ are open sets, why are the sets $A\cap V_1$ and $A\cap V_2$ open?

In a proof of a question I found the following sentence: "Since A⊂B, it follows that A∩V1 and A∩V2 are open in (A,d)." Why does it hold? I understand that if $x\in A\cap V_1\Rightarrow x\in A \...
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0answers
21 views

Graph of the composition of a continuous function and a correspondence with a graph homeomorphic to its domain

Let $\phi: \mathbb{R}^m \leadsto \mathbb{R}^n $ be an upper hemi-continuous correspondence, $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function. If the graph of $\phi$, $\{(x,y) \in \mathbb{R}^...
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1answer
34 views

Is it true that every totally bounded set in a metric space is compact?

Every compact set is totally bounded, but can we say that every totally bounded set is compact? I'm a beginner in metric space. My thinking is that a totally bounded set behaves like a finite set and ...
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28 views

Sharkovsky's Theorem and Triangular Functions

I'm trying to prove that Sharkovsky's Theorem Let $\vartriangleleft$ denote the Sharkovsky ordering given (informally) by $\underbrace{1\vartriangleleft 2 \vartriangleleft 4\vartriangleleft 8\...
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Is there a $D$- chain between two point of a connected component in an uniform space?

A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map of the space into a discrete space is a constant map also a topological space is connected if and only if ...
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2answers
44 views

Why is my proof that $\mathbb R$ is disconnected wrong?

The definition of connectedness in my notes is: A topological space $X$ is connected if there does not exist a pair of non empty subsets $U$, $V$ such that $U\cap V=\emptyset$ and $U\cup V=X$. ...
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3answers
60 views

Is the two-point compactification the second-smallest compactification?

We know that the Alexandroff one-point compactification of $\mathbb{R}$ is in a precise sense its smallest Hausdorff compactification. Is the two-point compactification of $\mathbb{R}$, in a precise ...
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1answer
38 views

What happens when every dense set is open?

I am supposed to prove or disprove the following claim: If in space $(X, \mathcal{O})$ every dense set is open, then $(X, \mathcal{O})$ is not $T_2$-space. I tried taking arbitrary $x, y \in X$ ...
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1answer
28 views

Irreducible closed subset

The dimension of a topological space $X$ is defined as the supremum of all integers $n\ge0$ for which there is a strictly increasing chain of $n+1$ irreducible closed subsets of $X$. Thus is the ...
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1answer
13 views

union of finite bounded set and uniformly bounded set is bounded

Let $A, G \subset C ([a, b])$, $G = \{g_1, g_2, ..., g_m\}$ (finite set). Prove that if: i) $A || .. ||$ $\infty$-bounded then $A \cup G$ too. ii) $A$ equicontinuous in $x_o$ then $A \cup G$ also. ...
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1answer
24 views

A cover of Locally connected space with certain compactness property

Suppose $X$ is a locally connected Hausdorff space. If $X$ is $\sigma$-compact and locally compact, is it always possible to find a countable set of precompact connected open sets $\{U_n\}$ (which ...
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2answers
113 views

A diifficulty in understanding a sentence in a paragraph in Guillemin and Pollack p.77

The paragraph is given below: But I have a difficulty in understanding the sentence starting in the forth line by "If we furthur ...." until its end, could anyone explain it for me please? thanks!
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2answers
46 views

If $int\overline{X} = int\overline{Y} = \emptyset$ then $int(\overline{X\cup Y}) = \emptyset$.

Let $X$ and $Y$ be subsets of a metric space. If $int\overline{X} = int\overline{Y} = \emptyset$ then $int(\overline{X\cup Y}) = \emptyset$. I know that $int(\overline{X\cup Y}) = int(\overline{X}\...
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1answer
48 views

Measurability of $\mathbb{R}^X$ in a compactification

Consider any (possibly infinite) set $X$. We fix on $\overline{\mathbb{R}}^X$ the $\sigma$-algebra product of the Borel $\sigma$-algebra on $\overline{\mathbb{R}}$ (where $\overline{\mathbb{R}}$ ...
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2answers
36 views

Counterexamples about function discontinuity.

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a function with a point $\textbf{x}\in\mathbb{R}^n$ of discontinuity. Is it possible that the image $f(O_{x_i})$, the image of an open ball (containing $...
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1answer
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1answer
25 views

In usual Euclidean metric on $\mathbb{R}^n$. Which of the following metric spaces X is complete [on hold]

For $X ⊂ \mathbb{R}^n$, consider $X$ as a metric space with metric induced by the usual Euclidean metric on $\mathbb{R}^n$. Which of the following metric spaces X is complete? $A. X = \mathbb{Z} \...
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2answers
52 views

What is the mathematical word to describe when two objects can be transformed to each other like Klein bottle and torus?

In mathematics how does one say that two objects like the Klein bottle and a torus can be transformed into each other and are the same thing in some sense?
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1answer
40 views

Subspace topology and product topology

Let $\tau$ be a topology on a topological space $X \times Y$ which is not a product topology. Consider the subspace topology $\tau_X$ and $\tau_Y$ induced by the topology $\tau$. My question is would ...
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0answers
48 views

Homeomorphism in compact two dimensional manifold, periodic points, and Euler Characteristic.

I want to prove that if a homeomorphism (a continuous bijection with continuous inverse) in a two dimensional manifold doesn't have a periodic point, then the Euler Characteristc of the manifold is ...
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1answer
17 views

How to find a completion for a metric space (For instance, support compact continuous real functions)

A completion of a metric space $(M,d)$ is a complete metric space $(M^*,d^*)$ such that $(M,d)$ is a dense subspace of $(M^*,d^*)$. I understand this, but how do I explicitely find a completion of a ...
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3answers
24 views

Union of closures

Let $X$ be a topological space $\mathscr{ B}$ be a collection of subsets of $X$. Show that $\overline{ \bigcup \limits_{\alpha \in \mathscr{B}} B_\alpha} \subset \bigcup \limits_{\alpha \in \mathscr{B}...
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1answer
61 views

Is this conjecture about the boundary of a surface correct?

I came up with an intuitive conjecture about boundaries of surfaces based on the idea that at a boundary point we can wrap a string across the edge, and the two halves of the string (on opposite sides ...
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2answers
36 views

What is an infinite subset of a compact set?

I am attempting to work on the following proof: If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$. I know that that this proof has been answered here already, but ...
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1answer
43 views

If $X$ is an infinite set, then $X$ with the discrete topology is not compact

I am new to compact sets, and I had some hard time trying to solve this: Let $X$ be an infinite set; $T$ the discrete topology on $X$ .Prove that $(X,T)$ is not compact. I know that I may consider ...