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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Topological Groups and the Mapping Class Group

I am currently studying mapping class groups. In particular, I am looking at a relation between the group of topological automorphisms of a topological group (i.e. group automorphisms which are also ...
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36 views

Correct definition of a regular covering without global connectedness hypotheses

Let $p:Y\to X$ be a covering map of topological spaces where $X$ is assumed to be locally path connected (and hence the same is true of $Y$) but neither $X$ nor $Y$ is assumed to be connected. In this ...
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81 views

In a complete metric space $(X,\rho)$, show that if $E$ and $X\setminus E$ are dense, then at most one of them is a countable union of closed sets.

The problem statement is in the title. I approached this proof using contradiction. My attempt was: Suppose that both $E$ and $X\setminus E$ are dense and that both are a countable union of closed ...
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104 views

Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
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20 views

Is there a theorem relates continuity of x-section&y-section of a function and continuity(measurability) of a function itself?

Let $(X,\tau),(Y,T),(Z,O)$be topological spaces. Let $f:X\times Y\rightarrow Z$ be a function. Let $f_x,f^y$ denote $x,y$-section of $f$ respectively. Let's assume $f_x,f^y$ are continuous for all $...
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236 views

A follow up question : compact sets included in a full, empty interior compact

This is a follow up question to When is a compact of the plane included in a connected compact with empty interior? I realized that I didn't quite correctly formulate the question I had in mind, ...
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50 views

Definition of Lebesgue number

I cannot understand the definition of Lebesgue number. Definition: Let $R$ be an open cover for a metric space $M$. A number $e>0$ is called a Lebesgue number if for all $x \in M$ there exist $U(...
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101 views

Difference between Metric Space and Topological Space

I am reading Chapter 11 of Real Analysis written by Royden and Patrick (4th). It says "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise ...
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54 views

Find Group Action

I'm asked to find a group action G on the unit cylinder C such that $C/G$ is homeomorphic to the torus. Would $\pm1 \cdot (x,y,z) = (x,y,\pm z)$ work? The only problem here is that G should only act ...
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307 views

Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
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43 views

When is a locally metrizable space actually metrizable?

A topological space is metrizable when there is a metric that induces the topology. A topological space is locally metrizable when every point has some neighbourhood that is metrizable. According to ...
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49 views

What are the names for the structures obtained when we drop some topological space axioms?

Motivation: If I start with the group axioms and drop the requirement that I have inverses, I get the monoid axioms. If I proceed to drop the requirement that I have an identity, I get the semigroup ...
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51 views

When can we say that each open set in $U_1 \subset X$ contains another open set $U_2 \subset U_1$ s.t. $U_2 \neq U_1$

As written in the title . Given a topological space $X$, what are the conditions required if we want to be able to say that each nonempty open set $U_1 \subset X$ contains another nonempty open set $...
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40 views

Compactness in topology of uniform conergence (of functions and all their derivatives) on compact subsets of (0,\infty)

I am trying to understand an example in the book "Lectures on Choquet's Theorem" (R.R. Phelps). My question is: Given the space of real valued infinitely differentiable functions on $(0, \infty)$ ...
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91 views

property of a two subset of the space $X=C[0,1]$ with its usual 'sup-norm' topology

Consider the space $X=C[0,1]$ with its usual 'sup-norm' topology.Let $S$={$f \in X:\int_0^1{f(t)dt \ne 0}$} $S_1$={$f \in X:\int_0^1{f(t)dt = 0}$} $1$.Then $S$ is (a) open , (b) dense in X , ...
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115 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ \...
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119 views

Proof of “invariance of domain theorem”

How can I get a proof of "invariance of domain theorem"? Of course, I would like to know brief explanation. Now, this statement is here ; http://en.m.wikipedia.org/wiki/Invariance_of_domain . Thank ...
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104 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ \...
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72 views

When is a function space a Fréchet space?

Let $Q$ be a space of indices, and let $(V, |\cdot|)$ be a Banach space of values. Define the function space $X = C(Q,V)$, and equip it with the topology generated by seminorms $\|x\|_D := \sup_{d \in ...
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59 views

Morse-Smale Complex, boundary on the number of segments by the number of critical points.

I am looking for a known upper bound on the number of monotone regions of a Morse function by the number of its critical points in the interior of the manifold and on its boundary. Here I try to ...
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44 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
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297 views

A domain on a sphere is simply connected if and only if its complement is connected

I think the statement that a domain (open connected set) in a sphere is simply connected if and only if its complement is connected is a standard result. But how can one prove it? Is it possible to ...
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130 views

Locally / Strongly connected product spaces

I have to give necessary and sufficient conditions s.t. $ X=\prod_i X_i$ is locally / strongly connected, where $(X_i,\tau_i)$ are non-empty top. spaces. First locally: Assume all $X_i$ are locally ...
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122 views

Set of limit points of S is closed in a metric space X

A point $x \in X$ is a limit point of a subset S of X, if every ball $B(x;\varepsilon)$ contains infinitely many points of S. Show that x is a limit point of S iff there is a sequence {$x_{j}$} in ...
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86 views

Exploiting the compactness of the unit circle to prove the following proposition.

I am trying to prove that a locally convex topological vector space is equivalent to a semi-normed topological vector space. I have worked through the proof but I am unsure because of the following ...
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81 views
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160 views

Point set topology from an algebraic perspective?

I got this idea of viewing a topology as an operation on a ring of sets. Let $\mathcal R = (\mathcal P(X), \cap, \triangle)$ be a ring of sets. ($\triangle$ is the symmetric difference operation and $...
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31 views

A question on the classical Mrowka space

Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset ...
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149 views

hyperspace of a complete uniform space need not be complete

I want to know the counter example for: Hyperspace of an arbitrary complete uniform space need not be complete. The hyperspace of a uniform space $(X,\mathscr D)$ is obtained by forming the set $\...
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48 views

What are default topologies on $R^∞$ and $R^ω$?

To extend the original question Difference between $R^\infty$ and $R^\omega$: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology ...
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139 views

Existence of points in closed and bounded convex sets that cannot be expressed as convex combination of other elements of the set.

I have an intuition about convex, closed, bounded sets but I can't really find a way to prove whether it's right or wrong. Let $\Sigma$ be a convex set, that means, that given any $A,B \in \Sigma$, ...
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41 views

Knowing the existence of a fixed point set from an induced fundamental group automorphism

Let $L$ be a link in $S^{3} $ and $f_{ \phi } : \pi_{1} (S^{3} \backslash L ) \rightarrow \pi_{1} (S^{3} \backslash L )$ be induced from a periodic map $\phi $ of $S^{3} $, restricted to the ...
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72 views

Interpreting Finite Topologies

Taking the quotes Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topological space there exists an open set not containing another (...
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83 views

How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
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156 views

Continuity of functional

This is a follow up question to this previous one. Let $X\subset\mathbb{R}_+$, where $X$ is countable and we are considering the space $\mathbb{R}^X$ of functions $f:X\to\mathbb{R}$ with the topology ...
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49 views

What is the name for the topology where every point is in the boundary of an open set?

Is there a name for topological spaces in which every point is in the boundary of an open set?
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49 views

The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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123 views

Is the category of topological spaces coregular?

Everything is in the title. The category Top of topological spaces and continuous mappings is not regular, but is it coregular ? Furthermore, Top isn't cartesian closed, but does it satisfy the dual ...
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136 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
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90 views

Product varieties with the constructible topology

Let $k$ be an algebraically closed field and let $X\subseteq k^n$, $Y\subseteq k^m$ be two affine algebraic varieties. It is not difficult to find examples where the Zariski topology on the product ...
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83 views

From Jordan's Curve Theorem to Jordan-Schoenfliess theorem

I am trying to learn and understand proofs of classical theorems and successfully mastered a proof of JCT. (It was the well-known proof that uses Tietze Extension and Brouwer's fixed point theorem). ...
2
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118 views

Proof that a set $X \subset M$ is a Manifold

Let M be a manifold without boundary and let , $g:M\to \mathbb R$ have $0$ as a regular value. Than the set $X \subset M$ with $g(x) \ge 0$ is a smooth manifold with boundary equal to $g^{-1}(0)$. I ...
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186 views

Orientability as a topological property

Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such ...
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305 views

Construction of Lakes of Wada

At each step of the construction of Lakes of Wada we extend a lake (an open set in the open unit square) so that no point of the land (the complement of all the lakes) is farther than a given small ...
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58 views

Quotient of complete linearly topologized ring

The quotient of a complete metrizable group by a closed normal subgroup is always complete, but there are examples to show this need not be true for non-metrizable groups. Here complete means every ...
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58 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
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46 views

Is there a name for a set where any two elements are separated by a given distance?

I am curious if there is a name for such a set. Let $(M,d)$ be a metric space and $S$ a subset of $M$ for which there is some positive number $\delta$ such that for any two distinct elements in $S$, $...
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237 views

An exercise from the handbook of set-theoretic topology

This is an exercise from the handbook of set-theoretic topology (Exercise 13.3): Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact ...
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80 views

derivatives using epsilon-delta argument

I have this question below and am not sure if my approach is correct. Can anyone please advise me? Thanks. Question: Let $f:\mathbb{R}^n \to \mathbb{R}^m$ and suppose there is a positive constant $K$ ...
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58 views

Finding a sufficient condition for a set to be finitely decomposable into open sets..

Let us call a set in a topological space finitely decomposable set (FDS) iff it can be rewritten using the standard set operations $\cup$ and $\sim$ and only finitely many open sets. I'm looking for ...