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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1answer
65 views

Continuity of distance function without triangle inequality

We say that a continuous function $\rho : \mathbb{C}^n \to \mathbb{R}$ is a distance function if the following three conditions hold: 1.) $\rho \geq 0$ 2.) $\rho (z) =0$ iff $z=0$ 3.) $\rho(cz)= ...
32
votes
3answers
3k views

Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
2
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0answers
46 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
3
votes
1answer
29 views

Equivalence of relative and (reduced) homology for arbitrary pairs

I could not find my mistake in the following argument, though I know it is wrong. This is more like a "Q&A", since there is nothing to "prove" in the positive sense. Here it goes: For an ...
11
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4answers
350 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
1
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1answer
44 views

Continuous function between topological spaces

Let $ (X,\tau_{X}) $ and $ (Y,\tau_{Y}) $ be topological spaces and $ f:X\rightarrow Y $ be a function. My question is how to show if for each $ A\subseteq Y $ , $\overline{f^{-1}(A)}$ $ \subseteq ...
7
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3answers
318 views

Topological characterization of the closed interval $[0, 1]$.

I would like to learn purely topological characterizations of the closed real intervals (to justify the existence of algebraic topology). In particular, such a characterization should not use real ...
3
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1answer
45 views

Topology Book including specific aspects

I am looking for a basic book about Topology (maybe also a bit of Functional analysis but basically Topology) including the following points (in addition to the basic points): $\bullet$ Seminorms ...
2
votes
1answer
61 views

If $A$ is compact and connected, then is $\Bbb R^2\setminus A$ connected? [closed]

Let $A$ be a compact connected subset of the plane. Is $\Bbb R^2\setminus A$ connected? Why?
5
votes
4answers
1k views

On a topological proof of the infinitude of prime numbers.

There is a proof of the infiniteness of prime numbers using Topology. I was only informed of the existence of this proof. They say it's very elegant. One could show how this proof?
1
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0answers
69 views

Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
0
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2answers
32 views

Existence of a boundary point

I am not particularly well-versed in topology, so I wanted to check with you whether there exists a much simpler argument to prove the following statement or whether there are problems with my proof. ...
0
votes
1answer
33 views

Hausdorff or weaklly hausdorff may apply

Let $X$ be a topological space and suppose that there is a countable collection of open sets $$\mathbb{B}\{U_1,U_2,…\}$$ which is a basis for the topology of $X$. Let $A\subset X$ and let $x\in ...
8
votes
2answers
227 views

Showing the Sorgenfrey Line is Paracompact

The Sorgenfrey Line is $\mathbb R_/ = (\mathbb R, \tau_s)$ where $\tau_s$ is the topology on $\mathbb R$ with base $\{[a, b)\ |\ a, b \in \mathbb R\}$. I know how to show $\mathbb R_/$ is not locally ...
3
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2answers
1k views

need one counter example for sum of two closed set need not be closed

I know the proof that If A is compact and B closed then A+B is closed but would like to have an example where both are closed but not A+B.I am not able to figure out.
0
votes
1answer
48 views

Homeomorphism are equivalence relations, so what are the equivalence classes?

Homeomorphisms are equivalence relations, so what are the equivalence classes for two Topological spaces $T_1, T_2$? Intuitively it seems like we might have the following equivalence classes - ...
1
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1answer
38 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
4
votes
2answers
77 views

Formally show that the set of continuous functions is not measurable

Let $C(\mathbb{R})=\{ f:\mathbb{R}\to \mathbb{R} \colon \ f \text{ continuous}\}\subseteq \mathbb{R}^{\mathbb{R}} $. How to prove formally that $C(\mathbb{R}) \notin ...
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1answer
66 views

Locally compact groups

Let $S= G_1\bigcup G_2 $, where $G_1$ and $G_2$ are two groups. If $S$ is locally compact, is it true that either $G_1$ or $G_2$ is locally compact? Thank you.
0
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0answers
19 views

Showing that bd$A$=$\{\vec v\in\mathbb{R}^n| d(\vec u,\vec v)<r\}$

Problem: And $\beta_r(\vec u)\equiv \{\vec x\in\mathbb{R}^n| dist(\vec u,\vec x)<r\}$. I got the first part showint that Int $A$=$A$. Now I want to show that bd$A$=$\{\vec v\in\mathbb{R}^n| ...
2
votes
4answers
125 views

$[0, 1)$ and $S^1$ not homeomorphic?

Let $f:[0, 2\pi) \to S^1 = \{(x, y): x^2 + y^2 = 1\}$ be such that $f(t) \to (\cos t, \sin t)$ $f$ is a continuous bijection but it is NOT a homeomorphism. I suppose the only point of contention is ...
1
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2answers
49 views

Can any collection of open sets in $\mathbb{R}$ be covered by a countable subcollection?

Let $A$ be a collection of open sets in $\mathbb{R}$. is there a countable subcollection $G_i$ of $A$ such that $$\cup_{G\in A} G=\cup_{i=1}^\infty G_i$$ I guess there must be such subcollection, but ...
0
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1answer
44 views

Orthogonality on Banach spaces

I got a doubt with a proof in Brezis' Functional Analysis, theorem 2.16. It says Theorem 2.16: Let $G,L \subset E$ be two closed subspaces in a Banach space $E$. Then the following properties are ...
2
votes
1answer
119 views

Unique smallest and largest topology ideas

What does it mean to be a smallest topology of a set $X$. I would guess that it would be a topology of $X$ which has least number of elements and similarly for largest topology it would have to be ...
2
votes
2answers
28 views

Doubt about limit point

I came though this definition about limit point A point z is a limit point for a set A if every open set U containing z intersects A in a point other than z. I want to know can we change it to ...
0
votes
1answer
26 views

First countable spaces, enumerable neighborhood base system

If $X$ is a topological space then $X$ is first countable if for every $x \in X$ there exists a neighborhood base system $\mathcal{B}_x$ that is enumerable. Does this means that $$\mathcal{B}_x = \{ ...
2
votes
1answer
30 views

Continous surjective map from $S^1$ to $S^n$

Is there any continous surjective map from $S^1$ or $[0,1]$ onto $S^n$, for some $n\geq 2$. Thank you.
1
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2answers
54 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
2
votes
1answer
60 views

Is $[0,M]^\infty $ connected and separable space?

I know that $[0,M]\subset R_+ $ is connected, separable. Now, let us consider the infinite dimensional space $[0,M]^\infty $. I want to see whether this space in connected and separable. I think the ...
74
votes
18answers
7k views

How to distinguish walking on a sphere or on a torus?

Imagine that you're a flatlander walking in your world. How could you distinguish if the world is a sphere or a torus ? I can't see the difference from this point of view. If you are interested, this ...
14
votes
1answer
348 views

What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. The study of characteristic classes tells us that these ...
-1
votes
0answers
146 views

If $A$ is complete for $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology

If $A$ is complete for both $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology. (Matsumura, CRT, Exercise 8.1) How can I solve this problem? A is a ring ...
1
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0answers
26 views

The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid ...
5
votes
2answers
86 views

Explicit expression for homeomorphism and homotopy equivalence

In many cases of topology when one needs to show two spaces $X$ and $Y$ are homeomorphic or homotopy equivalent, one uses some description instead of constructing an explicit homeomorphism or homotopy ...
4
votes
1answer
78 views

Topological degree of a complex valued map defined over a circle

Given a continuous map $f \colon S^n \to S^n$, it induces a map $f_{*} \colon \tilde{H}_n(S^n) \to \tilde{H}_n(S^n)$ of the form $f_{*}(z)=k*z$, where $k$ is an integer. Define the degree of $f$ as ...
2
votes
0answers
55 views

Continuous function - unsure of statement that lacks rigour

I have the following statement in my Topology notes in a section on continuous functions - A polynomial of degree $n$ has at most $n$ roots. Thus $f^{-1}(b)$ is finite. This shows that ...
1
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1answer
26 views

Any polynomial function is continuous - what about a constant function?

I read that any polynomial function is continuous. I.e. If we have an open set $U$ in the range, $f^{-1}(U)$ will be open in the domain. Let $\mathbb{R}$ have the standard topology. Define $f: ...
2
votes
0answers
37 views

explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
0
votes
2answers
38 views

Is this a quotient map?

Consider the map $f \mapsto f(0)$ from $\mathcal C([0,1])$ into $\mathbb R.$ Here $\mathcal C([0,1])$ is the space of continuous real functions on $[0,1]$ with the usual sup metric. Show that this is ...
2
votes
1answer
28 views

the homeomorphisms betwen two spaces looks like broom

Let $Y=${$(x,x/n)\in \mathbb{R} \times \mathbb{R}: x\in [0,1],n \in \mathbb{N}$} and $X=\cup_{n\in \mathbb{N}}{[0,1]\times (n)}$ and $(0,n) R (0,m),\forall n,m \in \mathbb{N}$. Then does $X/R $ is ...
2
votes
1answer
54 views

can a compact set have infinite measure?

Can a compact set have infinite measure? It does not seem to violate the measure axioms. This is not true in the case of Lebesgue measure. So I am also wondering is there any clean cut condition for ...
2
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1answer
48 views

Can any “relevant” topological spaces be decomposed into an uncountable product?

Can any "relevant", as meaning generally useful topological spaces be decomposed into an uncountable product of other topological spaces with the product topology? Many thanks in advance.
2
votes
2answers
39 views

Is Lower limit topology $\mathbb{R}_l$ finer than the standard topology $\mathbb{R}$?

Is Lower limit topology $\mathbb{R}_l$ is finer than the standard topology $\mathbb{R}$? In Munkres' topology, it's stated that $\mathbb{R}_l$ is finer than $\mathbb{R}$. In the argument , he is ...
0
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3answers
67 views

How to Show that $S=\{(x,y)\in R^2:x>y^2\}$ is open

Show that $S=\{(x,y)\in R^2:x>y^2\}$ is open quite simple one. We need to choose $\epsilon$ for open balls: $D((x_0,y_o),\epsilon)\subset S$ ,$\forall x_o,y_o\in S$. we can take $\epsilon$ as the ...
0
votes
1answer
44 views

Are circulant matrices open

Are the set of positive definite symmetric circulant matrices open in the set of positive definite symmetric matrices?
4
votes
1answer
62 views

Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$ TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-)) $$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
1
vote
2answers
51 views

Sequence of compact sets

Let $(X,d)$ be a metric space and consider an increasing sequence $A_n$ of its subsets such that $A = \bigcup_n A_n$ is compact. Can it happen that $A\setminus A_n$ is compact for all finite $n$?
0
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1answer
32 views

Proof of an open set or closed set

I'm struggling on a proof that I can't proof correctly. Let $A=\mathbb{Z}$, $B=\{n-\frac{1}{2n} | n \in \mathbb{N}*\}$ I could prove easily that A is a closed set and B as well : $\overline A =$ ...
2
votes
1answer
34 views

Product of Connected Spaces (2)

If $Y$ and $Z$ are connected, is $Y \times Z$ path connected? I cannot find a counter example. Some help please.
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0answers
25 views

What does it mean to say “Resolving intersections”

Consider a surface (with boundary) $S$ with marked points on the boundary such that we may may triangulate the surface. Call a line joining two marked points in a triangulation an arc. Consider a ...