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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8
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1answer
156 views

Are two spaces manifolds if their product is a manifold? [duplicate]

Possible Duplicate: Decomposition of a manifold For topological spaces $X,Y$, if their product space $X \times Y$ is a manifold, is it necessarily that $X,Y$ are manifolds?
0
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1answer
57 views

In an N-dimensional space filled with points, systematically find the point with highest spearmans correlation to a given-point

I asked a question exactly like this a while ago, so I do not know if it is appropriate to ask pretty much the same question with a single tweak. For the record, my first question is In an ...
1
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0answers
67 views

Adjoint to the Hom functor in Boolean rigs

What I wanna ask is about analogies to the tensor product for commutative boolean rings. What I mean by commutative boolean ring is set with two operations, + and *, and two identities, 0 and 1, as is ...
2
votes
1answer
152 views

In an N-dimensional space filled with points, systematically find the closest point to a specified point

This is my first post on the mathematics stack exchange site. If I am posting this question on the wrong site, I apologize and please let me know so I can delete it. I am a programmer, and one thing ...
2
votes
2answers
1k views

Limit point and interior point

Is any interior point also a limit point? Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a ...
4
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5answers
765 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
1
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1answer
164 views

Are there any synonyms of “pair of pants” in topology?

I used to know a term for pair of pants, but perhaps there is none. It looks like this also.
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1answer
51 views

How to show the subset $Y$ is closed discrete?

The example is following: Let $Y=\{(0,y):y \in R\}$. Let $E \subset R^2$, i.e., the subset of the real plane, and $E=Y\cup \{ (\frac 1n, \frac k{n^2}): n\in Z^+, k \in Z\}$. The topology on $E$ is ...
5
votes
1answer
403 views

Definition of Cantor Set without AC

You can see the original text that i thought AC is used here; From Walter Rudin: Principles of Mathematical Analysis, 3rd ed., ISBN 0-07-054235-X, p.41-42. 2.44 The Cantor set The set which we ...
2
votes
3answers
286 views

Zariski topology on prime $\mathrm{Spec}$ of a ring $R$

Let $R$ be a commutative unital ring. Let $\mathrm{Spec}(R) = \{ \mathfrak p \subset R \mid \mathfrak p \text{ a prime ideal of } R \}$. We define a set $C$ to be closed in this space if and only if ...
1
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0answers
84 views

Show a certain subspace of $\ell^p$ is compact. [duplicate]

Possible Duplicate: How to show that this set is compact in $\ell^2$ Here's my problem: Let $p \geq 1$ and let $(r_k)_{k=1}^\infty$ be a sequence in $\mathbb{R}$ such that $r_k > 0$ for ...
5
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3answers
152 views

The metrizable space may be not locally compact

My text book said: Not every metrizable space is locally compact. And it lists a counterexample as following: The subspace $Q=\{r: r=\frac pq; p,q \in Z\}$ of $R$ with usual topology, i.e., ...
5
votes
1answer
88 views

A question on the compact subset

This is an exercise from a topological book. Let $X$ is Hausdorff and $K$ is a compact subset of $X$. $\{U_i:i=1,2,...,k\}$ is the open sets of $X$ which covers $K$. How to prove that there exist ...
0
votes
1answer
261 views

Constructing a local nested base at a point

I am trying to prove the following: "Let $X$ be a first countable space and $x$ a member of $X$. Prove that there is a local nested basis $\{S_n\}_{n=1}^\infty$ at $x$." Since $X$ is first ...
0
votes
1answer
293 views

(ZF) Every nonempty perfect set in $\mathbb{R}^k$ is uncountable.

This is the part of proof in Rudin PMA p.41 Let $P(\subset \mathbb{R})$ be a perfect set. Since $P$ has limit points, $P$must be infinite. Suppose that $P$ is countable. Then, we can denote the ...
1
vote
2answers
252 views

Heine-Borel Theorem ($\mathbb{R}^k$) (in ZF)

Heine-Borel Theorem; If $E \subset \mathbb{R}^k$, then $E$ is compact iff $E$ is closed and bounded. I have proved 'closed and bounded⇒compact' and 'compact⇒bounded'. (There exists $r\in \mathbb{R}$ ...
0
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1answer
50 views

$I_n= (-n,n)×\cdots×(-n,n)$ in $\mathbb{R}^k$ is open

Let $I_n = (-n,n)×\cdots×(-n,n)$ in $\mathbb{R}^k$. ($n$ is a positive integer) Fix $x\in I_n$ Let $L=\{\min\{n-x(i),x(i)+n\} \mid i\in k\}$. Since $L$ is finite and is a subset of $\mathbb{R}$, it ...
2
votes
1answer
177 views

Open mapping of the unit ball into itself

Does there exist a continuous open function $f:B^n\to B^n$ which is not injective? (Here $B^n\subseteq\mathbb{R}^n$ is the open unit ball)
1
vote
1answer
123 views

On the intersection of closed sets

In a book on beginning measure theory, the following statement is made: "It is clear that any intersections and finite unions of closed sets are closed." However the intersection of two disjoint ...
3
votes
4answers
884 views

Every compact subset must be closed?

This is an exercise from a topological book. In $T_1$ space, every compact subset must be closed? For any two compact subset, their intersection must be compact? Thanks for any help:)
0
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0answers
426 views

Every k-cell is compact (in ZF)

This is the part of proof on my book. Let $I$ be a k-cell. Then for every $x\in I$ and $j\in k$, $a_j ≦ x(j) ≦ b_j$ for some $a_j, b_j \in \mathbb{R}$. Let {$G_{\alpha}$} be an open cover of I and ...
2
votes
2answers
120 views

proving “$C^1([−1,1])$ is dense in the given space with given norm”

Define $$E = \left \{ f \in W^{1,2} (-1,1) \; | \; \| f \|_E := \left( \int_{-1}^1 (1-x^2 ) | f' (x) |^2 dx + \int_{-1}^1 | f(x) |^2 dx \right)^{\frac{1}{2}} < \infty \right \}.$$ Then how can I ...
3
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0answers
258 views

Sequential compactness vs. countable compactness

Firstly, I'll give the definitions of sequential compactness and countable compactness. Sequential compactness: If $X$ is a Hausdorff space and every sequence of points of $X$ has a convergent ...
5
votes
0answers
181 views

Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
3
votes
3answers
372 views

Spectrum of a field

Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals. Thanks a ...
1
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1answer
346 views

Compact Connected Set in $R^2$ and Connected Component

Let X = compact connected set in $R^2$. Let $X^c$ be its complement. Am I right to say this: The number of connected components of $X^c$ roughly refers to the number of "holes" in $X$. So, I can ...
9
votes
5answers
905 views

The definition of metric space,topological space

I have read some books in analysis,all of them define metric space,topological space or vector space directly,without any reason. Therefore, I want to know the background of the definition, the ...
0
votes
3answers
105 views

Image of compacta under a continuous map

There is a well-known result in topology, Any continuous bijection from a compact topological space to a Hausdorff space is a homeomorphism. I was wondering whethet the following (slightly weaker) ...
3
votes
1answer
633 views

The relation between a metric space $(X,d)$ and the topological space that arises from it.

Consider the topological space $(\Bbb R,\mathfrak I)$ that arises from the metric space $(\Bbb R,d)$, with $d(x,y)=|x-y|$. I want to prove that $\partial(a,b)=\partial[a,b]=\{a,b\}$. I have that ...
1
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1answer
144 views

Uniform convergence of functions, Spring 2002

The question I have in mind is (see here, page 60, the solution is at page 297): Assume $f_{n}$ is a sequence of functions from a metric space $X$ to $Y$. Suppose $f_{n}\rightarrow f$ uniformly and ...
5
votes
2answers
481 views

$p$-adic completion of integers

I'm trying to do the following exercise: Let $p$ be a prime and for $n\geq 1$ let $\alpha_n :\mathbb Z/p \mathbb Z \to \mathbb Z/p^n \mathbb Z$ be the injection of abelian groups given by $1 \mapsto ...
4
votes
1answer
154 views

If $p$ is an element of $\overline E$ but not a limit point of $E$, then why is there a $p' \in E$ such that $d(p, p') < \varepsilon$?

I don't understand one of the steps of the proof of Theorem 3.10(a) in Baby Rudin. Here's the theorem and the proof up to where I'm stuck: Relevant Definitions The closure of the subset $E$ of some ...
2
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2answers
85 views

complex torus has topological genus one

could any one give me a hint how to show a complex torus has topological genus one by constructing an explicit homeomorphism to $S^1\times S^1$? Complex Torus: $\mathbb{C}/L$, where ...
1
vote
1answer
70 views

“Base for a neighborhood system at a point” vs. “base at a point”

For a topological space X do the terms "base for a neighbourhood system at a point" and "base at a point" have the same meaning?
0
votes
1answer
268 views

Uniform Convergence in Uniform Norm Topology

The resulting metric topology corresponding to the norm given by: $\|f\| = \sup\limits_{x\in X} | f(x) |$ on $C^*(X)$ is called the uniform norm topology on $C^*(X)$. Show that, in uniform norm ...
0
votes
1answer
86 views

Open relative and choice

I'm using ZF as my axiom system. Let $X$ be a metric space and $K\subset Y \subset X$. Suppose $K$ is compact relative to $X$. Let $\{V_a\mid a\in I\}$ be a family of open sets relative to $Y$ such ...
2
votes
1answer
156 views

Counterexample of Compactness

Let $X$ be a metric space and $E\subset X$. Let {$G_i$} be an open cover of $E$ For every open cover {$G_i$}, there exists a finite subcover {$G_{i_n}$} of $E$ such that $G_{i_n} \in${$G_i$}. For ...
0
votes
1answer
357 views

Definition of 'closed relative to'

I know what's the definition of 'open relative'. I googled 'close relative', but i couldn't find a definition of it. How come every metric space $X$ is close relative to $X$? If $p$ is a limit point ...
3
votes
2answers
111 views

map from $\mathbb{C}$ to $\mathbb{C}/L$ is open map?

Let $w_1,w_2\in\mathbb{C}$ be linearly independent vectors and let$$L=\{m_1w_1+m_2w_2:m_1,m_2\in\mathbb{Z}.\}$$ How does one show that the projection map $\pi:\mathbb{C}\rightarrow\mathbb{C}/L$ is ...
0
votes
1answer
92 views

Algebraic topology involved in the 1/4-pinched sphere theorem?

Can anyone familiar with this theorem and its proof let me know how much algebraic topology is involved, and where specifically? I am familiar with a lot of differential geometry, but not many of the ...
6
votes
1answer
142 views

Sequences, subsequences, and continuity of functions

It's been a few years since I studied point-set topology, and I'm a bit rusty on the basics. Would appreciate help with the following question. Suppose $f:X\rightarrow Y$ is a map between two ...
6
votes
1answer
217 views

Further examples of 1-transitive but not 2-transitive groups of self-homeomorphisms

In a question from last week, I was searching for "nice" spaces on which the group of self-homeomorphisms acted $1$-transitively but not $2$-transitively on the space. In that case, I had the ...
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2answers
113 views

Segment of $\mathbb{R}^2$?

I don't understand this sentence; The segment $(a,b)$ can be regarded as both a subset of $\mathbb{R}^2$ and an open subset of $\mathbb{R}^1$. If $(a,b)$ is a subset of $\mathbb{R}^2$, it is not ...
21
votes
2answers
593 views

The prime spectrum of a Dedekind Domain

Let $A$ be a Dedekind Domain, let $X = \operatorname{Spec}(A)$. Are all open sets in $X$ basic open sets? Thinking about the Zariski topology (in the classical sense) of a non-singular affine curve, ...
4
votes
1answer
184 views

Being second countable is invariant under perfect mapping

Firstly, I will give the definition of perfect mapping: Let $f$ be a closed mapping from a topological space $X$ to another topological space $Y$. We call it a perfect mapping if for every point ...
3
votes
2answers
220 views

If a subset of $\mathbb{R}$ is closed and bounded with respect to a metric equivalent to the Euclidean metric, must it be compact?

Two different metrics $d$ and $\hat d$ in a space $X$ are said to be equivalent iff the topologies generated by them are the same, in other words $U\subseteq X$ is $d$-open iff it is $\hat d$-open. ...
4
votes
1answer
133 views

Continuous Actions and Homomorphisms

I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of ...
3
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3answers
378 views

Boundary of product manifolds such as $S^2 \times \mathbb R$

Simple question but I am confused. What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$? What would be the general way to evaluate the boundary of a product manifold? Thanks for the ...
1
vote
1answer
81 views

Pseudocompact spaces

Suppose we have a pseudocompact, Hausdorff space $L$ (pseudocompact means that each continuous function $f\colon L\to \mathbb{R}$ is bounded). Consider the space $C(L)$ of continuous real-valued ...
2
votes
2answers
226 views

A question on a quotient of Alexandroff's double segment space

Does anybody know the Alexandroff's double segment space? References would be very welcome. I will try to describe it here: Alexandroff's double segment space: Suppose $X = C_1 \cup C_2$, where ...