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; ...

learn more… | top users | synonyms (3)

2
votes
2answers
30 views

Why one third belongs to Cantor set?

We know that all numbers that belong to Cantor set have a ternary representation with only 0's and 2's but for example $\frac{1}{3}=(0.1)_3$ and $\frac{1}{3}$ belongs to Cantor set. I don't understand ...
2
votes
0answers
16 views

Argument verification

Determine all of the accumulation points of the following sets in $R^1$ and decide whether the sets are open or closed or neither. e)All the numbers of the form $2^{-n} + 5^{-m}: (m,n = ...
0
votes
1answer
23 views

Question on proving quotient space is homeomorphic to circle

I am new to quotient spaces and was given this in class. I really have no idea how to solve. I tried one approach that my teacher said to be incorrect so I'd really appreciate the help on this. I am ...
1
vote
1answer
17 views

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ ...
3
votes
2answers
46 views

Prove that $U-f(U)$ is an open set.

Let $(X,d)$ be a compact metric space. Let $f:X\to X$ be continuous. Fix a point $x_0\in X$, and assume that $d(f(x),x_0)\geq 1$ whenever $x\in X$ is such that $d(x,x_0)=1$. Prove that $U\setminus ...
3
votes
1answer
18 views

Question on complete spaces, longer, more specific question.

Let $S \subset C^2[0,1]$ (set of two times differentiable functions $f(x)$ on $[0,1]$) which satisfy the following: $$\int_0^1 f(x)\,dx\leq3$$ Question is $(S,d)$ is a complete metric space, ...
0
votes
0answers
13 views

Fraction of Lipschitz functions among absolutely continuous ones

Is it true that the space of Lipschitz functions on $S^1$ is a $G_\delta$ subset of the space of absolutely continuous functions on $S^1$? In which topologies ($L^p$, uniform, $C^k$, etc) it is true? ...
5
votes
1answer
54 views

Recommendation for books on topology (light reads) [on hold]

Are there any books on topology which can be read without having to do any exercises and look up definitions every second line? Something to read while relaxing, and not meant to replace a textbook ...
3
votes
1answer
23 views

Compact set of real numbers with countably many limit points.

Construct a compact set of real numbers whose limit points form a countable set. My example: Let $E_1=\{1\}\cup \{1+1/n: n\in \mathbb{N}\},$ $E_2=\{1/2\}\cup \{1/2+1/n: n>2\},$ $E_3=\{1/3\}\cup ...
3
votes
1answer
29 views

Proving compactness by definition

Let $K\subset \mathbb{R}^1$ consists of $0$ and the numbers $1/n,$ for $n\in \mathbb{N}$. Prove that $K$ is compact directly from the definition (without using the Heine-Borel theorem) Proof: Let $K$ ...
0
votes
0answers
14 views

Is there an analog of of equilateral triangles of height = leg on a sphere for a tetrahedron and a 3-sphere?

I've run into that on a unit sphere, an equilateral triangle of leg length = pi/2 (pole to equator) will have a height of pi/2. Does this generalize upward that a tetrahedron can be mapped on the ...
0
votes
1answer
25 views

Find the interior points of the following set:

I can identify each element of the set $\mathbb{Q}$ $\cap$ $[0,1[$; however, I must confess that it is pretty hard for me to use the proper open ball for it. I would really appreciate your help. Have ...
2
votes
0answers
57 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
1
vote
1answer
42 views

$T:X \to Y$ bounded linear map and $X$ separable implies $Y$ is separable?

Let $T:X \to Y$ be a bounded linear map between Banach spaces. Suppose that $X$ is separable. Is it true that $Y$ has to be separable? I think yes, since the map is continuous it takes the ...
0
votes
0answers
41 views

Sum of the reciprocals topology

Let's define this topology in $\Bbb N$ (here $\Bbb N$ begins at $1$): $$K\subset\Bbb N\text{ is closed }\iff K=\Bbb N\;\text{ or }\;\sum_{n\in K}\frac1n\text{ converges}$$ I have worked some on it. ...
3
votes
1answer
68 views

Let $A ,B \subseteq \Bbb{R}^{k}$ and $A+B =\{a+b \mid a\in A, b\in B\}$then:

Let $A ,B \subseteq \Bbb{R}^k$ and $A+B =\{a+b \mid a\in A, b\in B\}$then: a)If $A,B$ be open then $A+B =\{a+b \mid a\in A, b\in B\}$ is open. b)If $A,B$ be connect then $A+B$ is connect? c)If ...
7
votes
0answers
78 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
1
vote
1answer
46 views

$f$ is continuous on $E$ if and only if its graph is compact.

This question may be asked before under different formulation, the original problem is Chapter 4, Exercise 7 of Rudin's text: The Principles of Mathematical Analysis: Problem: If $f$ is defined on ...
1
vote
0answers
23 views

Any online videos on a course taught from Munkres?

Are there any vidoes available on the Internet --- for watching online or for download --- of any (general) topology course taught using the book Topology by James R. Munkres, 2nd ed? If so, please ...
2
votes
0answers
57 views

Show that $X \cup \mathcal{E}(X)$ ( $X$ and the ends of $X$) is a compact Hausdorff space.

Let $X$ be a connected,locally connected, locally compact and Hausdorff space. Let $\mathcal{K}$ the family of compact subsets of $X$. An end , $\epsilon$ is a function: $\mathcal{K} \rightarrow X$ ...
0
votes
1answer
48 views

Topology on partially ordered set

Let $(X, \leq )$ be a partially ordered set. How would you define a topology on $X$ such that the closed sets are precisely the order-closed sets? Where $B \subset X$ is order-closed if ...
5
votes
1answer
47 views

Show the discrete topology is the only one larger than $\tau_l$

$(X,\le)$ is a partially ordered set, we define $U_l(x)=\{y\ |\ y\le x\}$, and $\tau_l$ is the topology generated by $\{U_l(x)\}$. We want to prove that the discrete topology is the only on that's ...
4
votes
2answers
52 views

Every neighborhood $N_r(x)$ in $\mathbb{R}^n$ is connected

I am working on an exercise in baby Rudin (Ex 2.20 in particular) and as part of that I am trying to show that any neighborhood in a metric space is connected. I've seen several differing definitions ...
0
votes
1answer
28 views

Finite closed covering of a bounded set in $\mathbb{R}^n$

My Attempt: I think here I can define the diameter of $A$ as follows since it is bounded. diam $A=\sup \{|x-y|: x, y \in \mathbb{R}^n\}$ So, I can take each $r_k$ as diam $A$. Am I on the ...
0
votes
1answer
83 views

Books for Ordinals and Cardinals

I am looking for a nice introductory book to read to learn and master ordinals and cardinals. Please help me!
0
votes
0answers
31 views

Does there exist a kuratowski set which is uncountable

A subset of a topological space is called the Kuratowski set if we can get 14 different sets by applying closures and complementation successively. I want to find a set which is uncountable and is a ...
1
vote
1answer
29 views

topological isomorphism between a group and product of its subgroups

I have stumbled upon the following question: Let $G$ be a $\sigma$-compact, locally compact Hausdorff group with $N$ and $H$ closed normal subgroups of $G$. Also $$N\cap H= \{e\}$$ and $$G=NH .$$ Then ...
2
votes
0answers
36 views

$gf$ closed with compact fibers $\implies f$ closed with compact fibers

Call a continuous function $\phi: A \to B$ universally closed if $\phi \times 1_T$ is closed for every topological space $T$. Exercise 3.6.13(d) of Ronnie Brown's Topology and Groupoids asks the ...
1
vote
1answer
12 views

Infinite set with discrete metric

Let $X$ be an infinite set. For $p\in X$ and $q\in X$, define $d(p,q)=1-\delta_{pq}$. Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are ...
0
votes
1answer
13 views

The set of all interior points. Set equality

$(\overline{E})^o=E^o$. Is this equality true? I proved that inclusion $E^o\subset (\overline{E})^0$ is true. But how to prove that inclusion $(\overline{E})^0\subset E^o$ is true or false? $E^o$ ...
1
vote
0answers
23 views

A basis $B$ for a topological space $X$ is a ring of sets iff $X\in B$

Let $X$ be a topological space with basis $B$, and suppose that $B$ has the following properties: $B$ consists of compact and dually compact$^\ast$ subsets. For every triple $U, V, W\in B$, we have ...
2
votes
1answer
35 views

Diameter of a 10-ball in a 10-box is larger than the side length of box?

I came across this idea in a lecture on elementary topology. While it makes sense algebraically, I'm hoping someone could shed some light on the way this is possible. So you begin with a square of ...
6
votes
8answers
1k views

What does it even mean to say 'preserve structure'? [duplicate]

Could somebody give a concrete example of a group structure being preserved in a isomorphism, et cetera? I always hear this 'preserve structure' thing. Ok, could somebody give me a rigorous definition ...
0
votes
1answer
38 views

A metric on the set of closed bounded subsets of a metric space

Define the distance from a point $p$ in a metric space $(X,d)$ to a subset $Y \subset X$ by $$d(p,Y) := \inf \{ d(p,y) : y \in Y \}.$$ For any $\varepsilon > 0$, define $$Y_\varepsilon := \{ x ...
0
votes
1answer
15 views

Codimension of the image of the polynomials subspace is infinite

Consider the interval $I=[0,1]$ and the Banach space $E$ of real continuous functions defined on $I$ ($E=\mathcal C_{\mathbb R}(I))$. $P \subset E$ is the subspace of polynomial functions (restricted ...
2
votes
2answers
32 views

Prob. 2 (e), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Open supersets and $\epsilon$-neighborhoods of closed noncompact sets

This question concerns exercise 2(e) from section 27 (p.177) in Munkres' Topology: Let $(X,d)$ be a metric space, and let $A$ be a non-empty subset of $X$. For any point $x \in X$, we ...
2
votes
1answer
48 views

Name for a continuous surjection such that $\operatorname{cl}(f^{-1}(A)) = f^{-1}(A') \implies \operatorname{cl}(A) = A'$

Consider a continuous surjective map $f \colon (X, \tau) \to (X', \tau')$ satisfying $$\operatorname{cl}(f^{-1}(A)) = f^{-1}(A') \implies \operatorname{cl}(A) = A'$$ for all $A, A' \in \wp(X')$ I ...
1
vote
2answers
68 views

How to negate: not a limit point (symbolic logic)

1.There are few I have seen here. $\forall N(x), \exists x'\in B, (x'\neq x\wedge x'\in E)$. $\forall N(x), \exists x'\in B, (x'\neq x\to x\not\in E)$. $\forall r>0, \exists x'\in N_r(x)\cap E, ...
0
votes
2answers
27 views

How to prove that E's limit point must be in E? (rudin)

How to prove that E's limit point must be in E? Thm 2.23 E's open iff E_c is closed. First, suppose E_c is closed. Choose x belongs to E. Then x doesn't belongs to E_c, and x is not a limit point ...
3
votes
1answer
43 views

Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
1
vote
1answer
66 views

Is the product topology the most finest topology you can give to the cartesian product and why?

I was reading about box and product topology which are given to Cartesian products . I want to know that is the product topology(excluding the box topology) the finest topology that I can give to a ...
3
votes
1answer
56 views

Homeomorphism between $S^2$ and $CP^1$ via uniqueness of quotient

I am trying to show that $S^2$ and $\mathbb{C}P^1$ are homeomorphic making use of the following result - see e.g. Jack Lee Introduction to topological manifolds. Let $Y \xrightarrow{\pi_1} X_1 $, $Y ...
2
votes
2answers
57 views

Borel set of $\mathbb R^n$ with $n > 1$

According to various sources, the Borel set over $\mathbb{R}^n$ can be defined in several equivalent ways: For instance, it can be defined as the smallest sigma-algebra containing every open set of ...
0
votes
0answers
55 views

proof with 3 quantifiers? [on hold]

Problems: $$∃x ∈ S \text{ s.t. } ∀y ∈ S, ∀z ∈ S, \text{ if } z > y, \text{ then } z ≥ x + y.$$ $$∀x ∈ S, ∃y ∈ S \text{ s.t. } ∀z ∈ S, \text{ if } z > y, \text{ then } z ≥ x + y.$$ What I have ...
0
votes
0answers
68 views

How to write a “set is open” or “set is closed” in a pure symbolic way with quantifiers? (FOL)

How to write a "set is open" or "set is closed" or "a set is open" in a pure symbolic way with quantifiers? And how to use pure symbol to prove "E is open iff its complement is closed"? and that ...
0
votes
1answer
28 views

support compact modulo subgroup

I am studying (co)-induced representations of topological groups and I came across the following situation: $G$ is a topological group, $H$ a closed subgroup and $f\colon G\to W$ a set-theoretic map, ...
3
votes
1answer
23 views

Lower Limit Topology Properties

I am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of ...
0
votes
0answers
25 views

Complex Projective Space as a Quotient of a Disc

I am reading Hatcher's book and I have a problem understading how the complex projective space $\mathbb CP^n$ can be realised as a quotient of $D^{2n}$ (page 7) Let me briefly outline his arguments ...
1
vote
1answer
30 views

Prove (in the example) that being homotopic depends on the range of the Homotopy

Question: Define $F : [0,1]\times [0,1] \rightarrow X$ by $F(x, t) = (cos(\pi x), (1 - 2t) sin(\pi x))$. Take a straight-line homotopy between $F(x, 0)$ and $F(x, 1)$. Show that they are ...
0
votes
0answers
47 views

Is the unit square a $2$-manifold in $\mathbb{R}^2$?

I'm using the following definition of a (smooth) manifold: It's from J.Munkres "Analysis on Manifolds". This is an exercise taken from this book: Is the unit square $[0,1]\times [0,1]$ a ...