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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2answers
24 views

Finite point set has limit points for general topological spaces?

I know that for the standard metric space on $\mathbb{R}$, a finite point set has no limit points. Does this also hold true for a general topological space? If not, is there a counter example?
3
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1answer
35 views

For $f :\mathbb R \to \mathbb R $, there exists an $(a,b)$, such that $f$ is bounded on a sequence with limit $x$, for all $x\in(a,b)$

I want to prove the following. Let $f : \mathbb R \mapsto \mathbb R $. Show that there exists an interval $(a,b) \in \mathbb R $ and $c >0 $: such that for any $x \in (a,b) $ there is a sequence ...
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2answers
41 views

Separated topological space

Let $\{x_1,x_2,\cdots,x_n\}$ $n$ different points $(n\in \mathbb{N}^*)$ from a separated topological space $(E,\tau).$ How to prove by induction that there exists $n$ neighborhoods ...
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1answer
48 views

Kelley's topology : A topological space X is compact iff each nest of closed non-void sets has a non-void intersection.

Recall that a nest is a family of sets which is linearly ordered by inclusion. This problem is from kelley's "general topology" problem 5.H. the necessity follows from the finite intersection ...
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1answer
85 views

Is $M$ is a compact manifold?

Let $M$ be a manifold of $m$--dimensional, and $M\subset \mathbb{R}^k$. Assume $m>n$. If every smooth function $f:M\longrightarrow \mathbb{R}^n$ has regular values that form an open subset of ...
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2answers
141 views

metric characterization for connectedness

Is there a metric characterization of connectedness? I'm looking for something like the following metric characterization of compactness: A metrizable topological space is compact if, and only if, ...
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1answer
39 views

Find two subsets $A$, $B$ of $\mathbb R$ such that $A^{a} = B^{a}$, $A\subset B$ but $\bar{A}\neq \bar{B}$.

Find two subsets $A$, $B$ of the real numbers with the usual topology such that $A^{a} = B^{a}$, $A\subset B$ but $\bar{A}\neq \bar{B}$. We know that the closure of $A$ is the subset of the closure ...
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1answer
64 views

Are there two subsets $A, B$ of $\mathbb R$ such that $\bar A = \bar B$, $A\subset B$; however, $A^a\ne B^a$.

Are there two subsets $A, B$ of $\mathbb R$ with the usual topology such that $\bar A = \bar B$, $A\subset B$; however, $A^a\ne B^a$? Here $A^a$ means the set of all limit points of $A$.
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2answers
155 views

Help with a proof that SO(n) is path-connected.

I've found lots of different proofs that SO(n) is path connected, but I'm trying to understand one I found on Stillwell's book "Naive Lie Theory". It's fairly informal and talks about paths in a very ...
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2answers
56 views

In metric space, $X$ is connected and $X\subset Y \subset \bar X$, prove that $Y$ is connected.

Question: $X$ is connected and $X\subset Y\subset\bar X$, prove that $Y$ is connected. This is one of my midterm questions this morning. I couldn't figure it out. But now I came up with this proof, ...
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1answer
48 views

Homology and Fundamental Group (Algebraic Topology - Allen Hatcher)

I have some questions regarding 2 parts of the theorem of this section: 1) Having $f = \sum_{i,j}(-1)^jn_i\tau_{ij}$, it pairs $\tau_{ij}$ with opposite signs (in any way I assume), and says that the ...
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1answer
54 views

About the proof in Topology Willard S. theorem 31.2

I have trouble to understand a few things in the proof: 1) Why is C connected? I understand that every $C_n$ is connected, and $C_{n+1}$ $\subseteq$ $C_n$. So if the intersection (that defines ...
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5answers
173 views

Why is Q dense in R?

Consider a topological space $S$ and an arbitrary subset $E$. Then if the closure of $E$ given by $\bar{E} = S$, we say that $E$ is dense in $S$. How can we prove that $\mathbb{Q}^{n}$ is dense in ...
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2answers
630 views

Is this set open or closed?

Is the set $(2, 3]$ open or closed, or both? I don't think it is open because no neighbourhood of the point 3 is totally contained in $(2, 3]$. Since it is not open, it must be closed. If it is closed ...
2
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1answer
60 views

Kelley's topology : A lemma about compactness and finite intersection property

I want to prove this problem by using the fact that a topological space is compact iff each net in it has a cluster point. Additionally, I believe that the finite intersection property may be ...
5
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1answer
164 views

Two versions of invariance of domain theorem?

While reading about Invariance of Domain Theorem, I noted that there are two common version of it, one saying that $f(U)$ is open, while another saying that $f$ is a homeomorphism (and sometimes both ...
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1answer
35 views

$R^\infty$ is separable

This question is from Introduction to Topology and Modern Analysis by Simmons. Question: It has to be proven that $R^\infty$ is separable where $R^\infty$ is defined as follows. The set of all ...
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0answers
23 views

Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
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1answer
34 views

Are there properties of vector space equipped with two norms?

I am interested in a vector space equipped with two norms$ \lvert \lvert \cdot \rvert \rvert$ and $ \lvert \lvert \cdot \rvert \rvert ^*$ satisfies that there is $M>0$ such that $ \lvert \lvert x ...
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0answers
72 views

Sets, Topology and Applying Cantor's Intersection Theorem

I am trying to solve the problem related to the Sierspinski triangle. The triangle is shown as follow. Let $S$ be the intersection of all the finite stages a). Show that $S$ is a nonempty compact ...
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1answer
35 views

contour which can be homeomorphic?

If I have a function $\phi:\mathbb{R^{2}}\rightarrow\mathbb{R}$ which is $C^{\infty}$ without critical points, can I assure that all the contour are homeomorphic?
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2answers
61 views

Is the set of matrices with rank at most $r$ closed? [duplicate]

The question is as follows: $\DeclareMathOperator{\rank}{rank}$ Is the set $S_r = \{A \in \Bbb R^{n \times n}: \rank(A) \leq r\}$ closed in $\Bbb R^{n \times n}$ in the Euclidean topology? I ...
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1answer
26 views

Correctness of reasoning about finiteness of degree of a covering map

Let $q$ be a covering $ q \colon \mathbb{R} P^{2n} \to X$, where $X$ is path-connected. Call $V_x$ the open nbhd of $x \in X$ given by the definition of covering map. We first note that $X$ must be ...
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3answers
115 views

Understanding the Co-finite Topology on R

I'm looking to gain a better understanding of how the cofinite topology applies to R. I know the definition for this topology but I'm specifically looking to find some properties such as the closure, ...
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1answer
26 views

Is the closure of an open connected set polygonally connected?

In an arbitrary metric space, I know that connected does not imply polygonally connected. However, in $\mathbb{R}^n$, an open connected set is connected iff it is polygonally connected. Is the last ...
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1answer
58 views

Show that a subset of a given function space is totally bounded

STATEMENT: Define a norm, $N$, on $\mathcal{L}(X)$ by $$N(f)=||{f}||_\infty+L(f)$$ Let $D=\left\{f\in\mathcal{L}_b(X):N(f)\leq 1\right\}$. Prove that when $X$ is compact and $D$ is viewed as a subset ...
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1answer
104 views

Definition of continuum

In class the teacher gave us the definition of a continuum as a metric, connected and compact space, on the book General Topology by Willard says that is a connected, compact and Hausdorff space, and ...
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1answer
70 views

what does “closed subspace” in papers mean?

In many books and articles one finds sentences like this: "let $A$ be a closed subspace of ...". Now my question might be stupid, but I am always wondering what they mean by closed subspace? Is this ...
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2answers
31 views

Equivalance of norms

Let $X$ be the vector space of all real valued functions defined on $[0,1]$ having continuous first-order derivatives. How to show that the following norms are equivalent: $\|f\|_1 = |f(0)| + ...
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0answers
29 views

Cut sets in topology

As part of a problem sheet I have been asked to show that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ whenever $n \neq m$. When I first proved that $\mathbb{R}$ is not homeomorphic to ...
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1answer
32 views

How to justify that $d_{\mid\cdot\mid} ^{\alpha} (x,y) = |x-y|^{\alpha}$ (with $0<\alpha <1$) is a distance on $\mathbb{R}$

I know directly that : $\forall x,y \in \mathbb{R}$, $d_{\mid\cdot\mid}^\alpha (x,y)=0$ $\Leftrightarrow$ $x=y$ $\forall x,y \in \mathbb{R}$, $d_{\mid\cdot\mid}^\alpha (y,x) = ...
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1answer
49 views

Partitions of $[0,1]$

Trying to test my understanding of analysis, today I came up with two questions, that will probably look obvious (be patience, because I am self-thaught). Anyway, here they are: 1) Is possible to ...
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2answers
48 views

Topology, locally compact

Let $E$ be a separable Banach space. Is $E$ locally compact space? I'm looking for a counterexample to this assertion. If you know anything please let me know.
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1answer
45 views

complete compact open topology

Let $X$ denotes a path-connected and compact manifold and $PX$ its path-space (the set of continuous maps $\gamma: [0,1] \longrightarrow X$) topologized with the compact open topology. It is true that ...
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2answers
171 views

Closed sets with empty interior measure zero

Is the Lebesgue measure of a closed set with empty interior in $\mathbb{R}^{n}$ always zero? Trying to understand something in the math notes that I don't understand, and if the above is true, it ...
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0answers
21 views

Examples of compact subsets in topological spaces [duplicate]

I'm trying to find a topological space and a compact subset whose closure is not compact. It is an exercise in the text book Armstrong, Basic Topology, which I can not figure out. Any hint or answer ...
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4answers
51 views

How to find a bounded sequences with three sub-sequences that converge to three different limits.

The question is about finding a bounded sequence with three sub-sequences that converge to three different limits.
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1answer
73 views

$\Bbb R \times \Bbb R$ with a nonstandard topology

Let $\tau$ the topology on $\Bbb R \times \Bbb R$ generated by the collection of lines $y= 2x+k$ with $k\in \Bbb R$ (in the sense that the line $\{y=2x+k\}$ is a basic element for $\tau$) . Find ...
3
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1answer
86 views

Proof of $\Bbb R^n \not\eqsim \Bbb R$

Why is showing the same space but without a point is equivalent to show that the original spaces are not homemorphic? From what i could tell they just showed $\Bbb R - \{ x \} \not\eqsim \Bbb ...
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2answers
103 views

Understanding how connected sum of smooth surfaces is a surface

I have two smooth surfaces $M_1$ and $M_2$ I''m trying to understand how the connected sum $M_1 \mathop{\#} M_2$ is a smooth surface. I will write my understanding of the proof and then explain where ...
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2answers
62 views

How do I show using cut points that these two subsets of the plane are not homeomorphic?

How do I show using cut points or cut pairs that these 2 subsets are not homeomorphic? I cannot see any obvious cut points. I see that, in the first diagram the only cut pairs of type 1 when 1 ...
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2answers
56 views

Closure of this set is $\mathbb R^+$

Let $A$ be a subset of $\mathbb R^+$ which is not bounded. Prove that $\displaystyle \operatorname{cl}(\bigcup_{n\in\mathbb N^*}\frac{1}{n}A)=\mathbb R^+$ where $\frac{1}{n}A$ denotes ...
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1answer
64 views

How do I prove that a subset is closed in the topological space of $n \times n$-matrices.

Consider the topological space $M$ of $n \times n$ matrices over $\mathbb{R}$ equipped with the standard topology. Let $\mathcal{A} \subset M$ be the set of matrices such that $det(A) = 1$ for $ A ...
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2answers
84 views

Is $\{(x,y)|y=\sin(\frac{1}{x})\}\cup(0,1)$ connected on $R^2$? [duplicate]

Source of definition: http://mathworld.wolfram.com/ConnectedSet.html Definition: A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology ...
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3answers
604 views

Example of a countable open set of real numbers?

I am unable to think of a set that is both open and countable. While I can easily think of several that are not-closed and countable, finding explicitly open ones (besides the empty set) is proving to ...
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0answers
83 views

Homotopic attaching maps give Homotopy Equivalent spaces

I want to prove that if $f,g : S^{n-1} \to X$ are homotopic maps then the resulting spaces $X \cup_f D^n$ and $X \cup_g D^n$ are homotopy equivalent. I know this question has been asked before: ...
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0answers
20 views

Continuity definitions on non-compact subsets

At the top of the wikipedia articles on Hölder condition, Lipschitz Continuity and others, the chain of logic is as follows: On a compact subspace of a metric space: $$Continuously Differentiable ...
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1answer
24 views

Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?

I'm trying to construct a homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$. I'm pretty sure there is one. I've been trying to work geometrically : mapping $[0,1]\times[0,1]$ to ...
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2answers
82 views

Cantor Sets/nonempty/cardinality

Let $S_0=[0,1]$ and define every $S_k$ for $k\geq 1$ \begin{align*} S_1&=\left[0,\frac{1}{3}\right]\cup\left[\frac{2}{3}, 1\right],\\ S_2&=\left[0,\frac{1}{9}\right]\cup\left[\frac{2}{9}, ...
4
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1answer
56 views

Extension of the limit operator on $l^\infty$

Let $l^\infty = \{x\in \mathbb{R}^\mathbb{N}\colon \sup_{n\in \mathbb{N}}|x_n|<\infty\}$ and the subspace $C \subseteq l^\infty$ given by the convergent sequences. We consider the linear operator ...