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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Concordant maps and their $n$-th powers

Let $f:X\to Z$ and $g:Y\to Z$ continuous maps where $X$ is a subspace of $Y$ and $f=g\restriction _X$. Then $f$ and $g$ are called concordant if $f^{-1}(z)$ is a dense subset of $g^{-1}(z)$ for every ...
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46 views

$\sigma$-$\sigma$-compactness is $\sigma$-compactness?

I mean, if $X=\displaystyle\bigcup_{n\in\mathbb{N}}K_n$ where each $K_n$ is $\sigma$-compact, then $X$ is $\sigma$-compact? I'm not sure if a countable union of countable unions is still a countable ...
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34 views

Continuity of certain projections in the weak topology.

I'd like to prove that: Given a Hilbert space H and S a closed subespace, $S \subseteq H$, the projection $P_{S}:H \to S$ is continuous in the weak topology. I have tried the following. ...
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1answer
57 views

continuous of a function

Let $U$ be a non-empty open set in $R^2$ and let $f:U\to R$ be a function. Suppose that the first partial derivactives of $f,f_1,f_2$ are defined and bouned on all of $U$. Show that $f$ is continuous ...
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106 views

How to complete this particular proof of $cl(cl(A)) = cl(A)$ for a subset $A$ of topological space $X$?

I want to prove that $cl(cl(A)) = cl(A)$ for a subset $A$ of topological space $X$. Given a subset $A$ of a topological space $X$, I have at hand the following two equivalent definitions about the ...
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30 views

disjoint compact subsets of $\ell^\infty(\mathbb{R})$

What would be some compact subsets of $\ell^\infty(\mathbb{R})$ which are disjoint? I know that the set of convergent sequences in $\mathbb{R}^n$ is one compact subset, but what would be another which ...
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1answer
75 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
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81 views

Connectedness in proximity spaces

Let $\delta$ be a proximity. A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$. ...
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75 views

What are epimorphisms in the category of Hausdorff spaces? [duplicate]

Let Haus be the category of Hausdorff spaces whose morphisms are continous maps. It seems that epimorphisms of Haus are those maps whose images are dense in the target spaces. How do you prove or ...
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29 views

Show that un in a compact topological space, any infinite set has some limit point. When does the reverse hold?

Show that un in a compact topological space, any infinite set has some limit point. When does the reverse hold? My attemp: I have done the proof but i dont know when the reverse hold. I have come up ...
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1answer
54 views

A statement on a set and its cluster

Let $X$ be a compact metric space and $f:X\rightarrow X$ be a homeomorphism. we define the orbit of a point $x$ as $\mathcal{O}(x)=\lbrace f^n(x): n\in\mathbb{Z}\rbrace$.let $\mu$ be the borel measure ...
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52 views

Resolvable spaces

a space $X$ is called a resolvable space if it is expressible as a union of two disjoint dense subsets. I want to find a resolvable but not lindelof space? Is there any example such a space?
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39 views

Sets that have the property of Baire

Can I say that a set $A$ has the property of Baire, if and only if it is of the form $A=(B \setminus C) \cup D$ where $B$ is regular open and $C,D$ are of first Category? Are there any other useful ...
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1answer
98 views

Spaces having a dense subset of isolated points

Is there anything known about spaces having a dense subset consisting of isolated points?
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44 views

A “complementary” topology

If $(X, \tau)$ is an Alexandrov topology then arbitrary intersection of open sets are open, and likewise arbibtrary unions of closed sets are closed, so we can define a topology $(X, \tau')$ as $$ U ...
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38 views

G-space decompositions preserved by equivariant maps?

Let $X,Y$ be topological $G$-spaces, with (left) $G$-invariant probability measures $\mu_X,\mu_Y$ respectively, and let $f:X \to Y$ be a surjective $G$-equivariant map preserving the measures, i.e. ...
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94 views

Brouwer's fixed point theorem on a sphere

Let $f: S^{2} \rightarrow S^{2} $ be a continuous map such that there exists a closed "disk" $D$ that is mapped to itself. Then will $f$ have a fixed point?
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100 views

Complement of set of all condensation point for an uncountable set of reals is at most countable.

Perfect Set: A set $E \subset X$ is said to be perfect if $E$ is closed in the metric space $(X,d)$ and every point of $E$ is a limit point of $E$. Condensation Point : A point $p \in X$ is said to ...
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36 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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53 views

A criterion for compact topological spaces

A book I'm reading uses the following argument, but I could't see how it works: Suppose $X$ is a topological space, $\mathcal{A}$ is a pre-basis for the topology (i.e. the topology on $X$ is the ...
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33 views

Cylinder as Fibre bundles

I have to show that the cylinder C is a fibre bundle over $S^1$ with fibre an open interval and I have to write a trivialization and the cocycles. I think that this is a trivial bundle, because I can ...
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30 views

When do continuous surjections have Borel sections?

It is known that whenever we have a continuous, surjective map $f\colon X\to Y$ between compact metrisable spaces, there is a Borel (even Baire class $1$) section $g\colon Y\to X$ (so that $f\circ ...
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25 views

Topology - Connected Images

Let X be a topological space and let Y = {0,1,2} have the D topology. Assume f: X$\rightarrow$Y is a continuous function. If A is a connected subset of X, what are the possible values of the image ...
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39 views

Topology - Projections

I'm pretty sure I have this right, but want to double check and make sure. Let $X_1$ = $X_2$ = $\mathbb{R}$ and let $p_1: X_1 \times X_2 \rightarrow X_1$ and $p_2: X_1 \times X_2 \rightarrow X_2$ be ...
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1answer
58 views

Abelianized fundamental group

Let $P$ be the projective plane and let $nP$ be the connected sum of $n$ copies of the projective plane. Show that the abelianized fundamental group $\pi_{1}(nP)/[\pi_1,\pi_1]$ is the direct sum of a ...
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57 views

Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
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53 views

Hausdorff and Quotient Spaces

Let $L$ be a subset of $\mathbb{R}^{2}$ and let $N = \mathbb{R}^{2}/L$ be the quotient space obtained by identifying all points in $L$ to a single point. I need to prove that $N$ is Hausdorff ...
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1answer
67 views

What can we say about the intersection of a clopen set and a connected set

Let $X$ be a metric space and let $E \subseteq X$ be a clopen subset. let $A \subseteq X$ be a connected subset. What can be said about $A \cap E$ when $A \cap E \neq \varnothing$? I believe that $E ...
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1answer
40 views

What is the closure of this set?

$\{a_n\}_{n=1}^{n=\infty}$ is a bounded sequence in $\mathbb{C}$ Is ${\overline{\{a_n\}}}=\{a_n\}$ ?
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105 views

Connection between sequences and filters in first countable spaces

It is generally known that the concept of sequences does not yield a satisfactory theory of convergence in arbitrary topological spaces. Instead one considers more general objects such as filters. ...
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22 views

Covering spaces and Automorphisms

I need to find for the groups $G$ a connected degree-4 cover $\hat{B}\rightarrow B$ such that Aut($\hat{B}\rightarrow B$) is isomorphic to $G$ $G \cong 1$ $G \cong \mathbb{Z}_{2}$ $G \cong ...
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1answer
68 views

for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.

Let $X$ and $Y$ be two topological spaces. Prove that for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous. I am stuck on the converse. ...
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1answer
51 views

Prove that set is arcwise-connected. Function $\mathbb{R}^3 \to \mathbb{R}^2$ with differential of rank $2$.

Let $f:\mathbb{R}^3 \to \mathbb{R}^2$ and assume that $0$ is regular value of $f$ (i.e. the differential of $f$ has rank $2$ at each point of $f^{-1}(0)$). Prove that $\mathbb{R}^3 \setminus ...
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2answers
90 views

Error in Engelking?

I've been trying to do some exercises in Engelking's General Topology text, and there's one that's causing me problems. I hope that someone here can clarify this for me. The exercise is (slightly ...
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49 views

$C(X,Y)$ complete

I want to prove that: $C(X,Y)$ is complete in the compact-open topology, when every component of $X$ is locally compact with a countable base, and $Y$ is a complete metric space. The proof I am ...
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27 views

Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
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94 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
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22 views

How do i prove an *infinite* product of connected spaces is connected? [duplicate]

Let $(X_i,\tau_i)_{i\in I}$ be a family of connected spaces. I couls prove this when $I$ is finite, but how do i prove this when $I$ is infinite?
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46 views

Construct a topological manifold which its open cover is locally finite but not globally

The whole question is like this: 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely many others, show that U is locally ...
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24 views

Self similar set and its measure

Prove: If $(K,\{f_i\}_{i=1}^N)$ is a self-similar set and $(\mu,\{\mu_i\}_{i=1}^N)$ is a self-similar measures, there is any arbitrary partition $\Lambda=\Lambda_a(r_1,\cdots,r_N)$ and ...
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30 views

Convergence of a Sequence of Functions

The context: verifying the group axioms for the fundamental group, specifically that every element has a unique inverse. Below is a non-example, and I am tasked with explaining $why$ it fails. Let ...
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61 views

What is the picture for $S^1\times S^2$?

Let M be a compact, connected, two-dimensional manifold such that $M = S^1\times S^2$. How should one picture M? The following is from Abraham and Marsden, Foundations of Mechanics: Let $M$ be ...
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1answer
42 views

Question about strong and norm convergence.

Maybe the answer to this question is so trivial that I can't see it: Why the strong convergence of operators (on an hilbert space) does not imply the norm convergence? Many books make this example: ...
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Count the number of topological sorts for each poset [duplicate]

Count the number of topological sorts for each partially ordered set $(A,|)$, where (a) $A = (3, 5, 7, 11, 13, 16, 17)$ (b) $A = (1, 3, 9, 27, 81, 243)$ That is, you have to find the number of ways ...
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2answers
98 views

Prove $\{(x,y) \in \mathbb R^2|x^2 + y^2 > 1 \}$ is not simply connected

I have literally no idea how to do this. My assignment question asks me to prove that $\{(x,y) \in \mathbb R^2|x^2 + y^2 > 1 \}$ and $\{(x,y) \in \mathbb R^2|x^2 + y^2 < 1 \}$ are homeomorphic ...
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76 views

What is an “essential loop”?

I'm a bit confused. Is an essential loop in a topological space $X$ just a loop $\alpha$, which is not-contractible (i.e. $[\alpha] \neq 0$ in the fundamental group of $X$), or is there something more ...
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1answer
22 views

When does a subspace have the same regular open algebra?

Given a topological space $X$ and a dense subspace $D$, I believe it's true that for a regular open set $U$ of $X$, $U \cap D$ is regular open in $D$. Note this induces a homomorphism between the ...
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1answer
47 views

Normal spaces in box and uniform topology

Is $\mathbb{R^\omega}$ normal in product topology? In the uniform topology? and In box topology? My attempt:I know it is normal in uniform topology because it is metrizable. I would gues that it is ...
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2answers
64 views

Topology - Composition of two isometric embeddings

Prove that the composition of two isometric embeddings is an isometric embedding and that the composition of two isometries is an isometry. I have been working on this problem for sometime, if ...
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75 views

Relative Interior and dense subsets

(Due to no answers, I also posted this question here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ ...