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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4
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1answer
203 views

Products of CW-complexes

I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using ...
3
votes
0answers
12 views

Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
0
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0answers
4 views

Finite Number of Partitions of Unity in a Compact Hausdorff Space

I'm working on this proof in Gamelin "Introduction to Topology" and I think I'm almost at the result, I'm just a little stuck with how to proceed. It is this. Let $X$ be a be compact Hausdorff space ...
2
votes
1answer
386 views

Show that a finite union of compact subspaces of a topological space $X$ is compact.

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify my proof or offer suggestions for improvement? Show that a finite ...
1
vote
1answer
24 views

Interior of A-closure, isolated points [duplicate]

I have been working on this question. Let A be an open set. Does Int(A-closure)= A? Here is answer #1: Let A = (0,1) union {2}, where A is a subset of [0,1] union {2}. Also, A-closure = [0,1] union ...
0
votes
1answer
18 views

To show that product $Z=X×Y$ in the product topology is a CW complex

I would like to prove the following: If $X,Y$ are CW complexes, and either $X$ or $Y$ is locally compact then the product $Z=X×Y$ in the product topology is a CW complex. (-see here) In order to ...
5
votes
1answer
27 views

What shapes, with boundary collapsed to a point, are homeomorphic to $S^n$?

Consider the following construction: Given a set $A\subseteq\Bbb R^n$, form the quotient space $A/\sim$ which identifies all the points on the boundary $\partial A$ (w.r.t $\Bbb R^n$). For which ...
3
votes
1answer
52 views

Is the surface of a donut distinguishable from pac-man's world?

Pac-man's world is topologically like the surface a donut. Pac-man's world is also locally flat. For example, the interior angles of a small triangle will always add up to 180 degrees. Conversely, ...
0
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2answers
45 views

Infinite product probability spaces

Does the infinite product of probability spaces always exist (using the sigma algebra that makes all projections measurable and providing a probability measure on this sigma algebra)? I always ...
1
vote
1answer
12 views

$\mathcal{T_B}$ is the intersection of all topologies containing $\mathcal{B}$

Let $\mathcal{B}$ be a basis on a set X, and let $\mathcal{T_B}$ be the topology it generates. Show that $\mathcal{T_B} =\bigcap \{ \mathcal{T} \subseteq P(X) \mathcal{T}$ is a topology on X and ...
0
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0answers
46 views

Urysohn's Lemma, Stone-Weierstrass

Let $X$ be a compact space. Show that the following statements are equivalent: a) $X$ is homeomorph to a compact subset of $\mathbb{R}^n$ b) There are functions $f_1,\dotso, f_n\in ...
1
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0answers
53 views

Impossible Covering Properties for Sets of Reals

I've been reading more about selection principles (covering properties) recently. Below is terminology. Adapting what B. Tsaban said in this article, we consider spaces $X$ which are (homeomorphic ...
2
votes
1answer
31 views

The Presentation Complex of $\mathbf Z\times \mathbf Z=\langle x, y|\ xyx^{-1}y^{-1}\rangle$ is the Torus

Let $G=\mathbf Z\times \mathbf Z$ and let $\langle x, y| \ xyx^{-1}y^{-1} \rangle$ be a presentation for $G$. In Example 1.46 of Hatcher's Algebraic Topology, the author mentions that the ...
2
votes
1answer
31 views

Are arithmetic operations open maps?

Viewing $+,-,\times , \div$ as function $R \times R \rightarrow R$, I 'proved' they are open maps by showing that for each function, the images of elements of basis for box topology (of form $(a,b) ...
0
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0answers
14 views

Neighbourhoods in compactly generated spaces

I ask this question, mainly motivated by a related unanswered question about the proof in Willard of the Arzela-Ascoli theorem. There it was assumed for the domain to be compactly generated, but the ...
0
votes
1answer
12 views

Countable dense set and basis in product spaces $\{0,1\}^{\mathbb{N}}$, $[0,1]^{\mathbb{N}}$ and $\mathbb{R}^{\mathbb{N}}$

For $\mathbb{R}^{\mathbb{N}}$ and $[0,1]^{\mathbb{N}}$ I know a set of sequences that are constant for almost all $n$ is dense, but how to make it countable? Changing the condition to equal $0$ for ...
0
votes
2answers
23 views

Proof that function on topological space is continuous if and only if 2 restrictions of it are

Topology such that function is continuous if and only if the restriction is. I've already seen this post but it didn't really help. The problem is the following: Let $X$ and $Y$ be topological ...
-3
votes
0answers
30 views

Given the following set, how do I determine the boundary? [on hold]

$$U = \left\{(x,y) \in \mathbb{R}^2: y=x^2, x\in (-1,1)\right\}$$ For space with taxicab metric.
1
vote
2answers
50 views

Show the “clock”and Euclidean metrics generate different topologies

I'm trying to teach my self topology. I wanted to find an example of a metric generating different topology. I came up with what a call "clock" metric, inspired by the modulo operation. Can anyone ...
0
votes
0answers
11 views

Compact-Open Topology for Space of C^{r} -sections

Given a smooth fibre bundle $\pi: X \rightarrow M$. What is the definition of compact open $C^{r}$-topology on the space of $\mathcal{C}^{r}$-sections?
10
votes
3answers
2k views

May a 'ball' that has been 'cut off' still be called a 'ball'?

Consider the metric subspace $[0,1] \subseteq \mathbb{R}$ with the metric defined in the usual sense, and the ball $B(0,1)$, defined to be the ball centred at $x=0$ with radius $1$. Now since only ...
5
votes
2answers
84 views

What is wrong with this proof that the identity map of $S^1$ is nullhomotopic?

I have read that the identity map of the unit circle $S^1$ is not nullhomotopic. In fact, I am very new to the subject, so I wonder what is wrong with the following reasoning (that seems to suggest ...
4
votes
1answer
723 views

When is Quotient Map a Covering Map

Group $G$ acts on topological space $X$. Also, $x,x'\in X$ not in the same orbit of $G$ have open $U$, $U'$ such that $g(U)\cap U'=\varnothing$ for all $g\in G$. I have shown that $X/G$ is ...
1
vote
1answer
41 views

Can you give me an example of a function that is either upper OR lower quasi-continuous but not both?

A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon >0$ and for each neighbourhood $U$ of $x$ there is a non-empty open set $G ...
2
votes
2answers
201 views

Continuity of sum/product using characteristic property of product topology

I'm self studying Lee's Introduction to Topological Manifolds, and I'm familiarising myself to the characteristic/universal property view of topology. One of the exercises in chapter 3 goes as ...
0
votes
0answers
25 views

The bases for the set of all functions f:[0,1]→[0,1]

Let $X = [0, 1]^{[0,1]}$, the set of all functions $f : [0, 1] \rightarrow [0, 1]$. Given a subset $A \subseteq [0, 1]$, let $U_A = \{ f \in X : f(x) = 0 \forall x \in A \}$ . Show that $B := \{U_A : ...
0
votes
0answers
21 views

Exploring the properties of the Srogenfrey Line

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
0
votes
0answers
19 views

Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
3
votes
1answer
34 views

How do I show that a contraction mapping in a metric space is continuous?

I start out by letting $V$ be an arbitrary open set in $X$. Then $$ f^{-1}(V) = \{x\in X\mid f(x) \in B_\epsilon(f(a))\}. $$ This can be re-written as: $$ f^{-1}(V) = \{x\in X\mid d(f(a), f(x)) < ...
0
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0answers
15 views

Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?

Here is Prob. 20 (a) in the book Topology by James R. Munkres, 2nd edition. Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of these topologies are the ...
0
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1answer
25 views

Confusion of classification of closed surfaces

I read that we can distinguish closed topological spaces without boundary up to homeomorphism by orientability and euler characteristic - is this correct? But what confuses me is that the Klein ...
2
votes
2answers
36 views
+50

Proof that a discrete space (with more than 1 element) is not connected

I'm reading this proof that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton ${x}$ and its ...
2
votes
1answer
25 views

A question about open subsets of the plane whose components are the interiors of circles.

We assume the Axiom of Choice. Let the term "Bubble Set" denote a proper non-empty open subset of the Euclidean plane P, all of whose components are the interiors of circles-i.e. open disks-which are ...
0
votes
1answer
20 views

How do I sketch a set and find its edge? How do I decide whether the following set in $\mathbb{R}^2$ is open, complete, limited, or compact?

How do I sketch a set and find its edge? How do I decide whether the following set in $\mathbb{R}^2$ is open, complete, limited, or compact? $$ A=\{(x,y)\in \mathbb{R}^2:\left | x \right |+\left | y ...
1
vote
1answer
23 views

A fundamental question about disjoint union topology

According to the following link : https://drexel28.wordpress.com/2010/04/02/disjoint-union-topology/ A disjoint union topology , the disjoint union is a collection of tuples of the form ($\ x$, $\ ...
0
votes
1answer
23 views

Describe the quotient space $Y=S'/((1,0)\sim(0,-1)\sim(0,1)\sim(-1,0))$

Describe the quotient space $Y=S'/((1,0)\sim(0,-1)\sim(0,1)\sim(-1,0))$ $S'$ be the unit circle in $R^2$ I just simply cant visualise this space also how do i check if its connected or compact? ...
1
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0answers
65 views

Prob. 4 (a), Sec. 20 in Munkres' TOPOLOGY, 2nd edition: Which of these functions are continuous in these topologies? [duplicate]

Let $\mathbb{R}^\omega$ denote the set of all infinite sequences of real numbers. Let the uniform metric $\tilde{\rho}$ on $\mathbb{R}^\omega$ be defined as follows: $$\tilde{\rho}(x,y) \colon= \sup ...
4
votes
1answer
68 views

Functorial modifications of a topology

Let $S$ be a set. Then there are only two ways to attach functorialy a topology $\mathcal{T(S)}$ to it: The discrete and the trivial topology. Functorial means in this case that all maps $f \colon S ...
0
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0answers
20 views

About the proof of a corollary of Arzela-Ascoli Theorem.

This is from Scheidemann, Complex Analysis. Theorem (Arzela-Ascoli): Let $K$ be a compact separable metric space, $E$ a finite-dimensional Banach space and $(f_j)_{j\in\mathbb{N}}\subseteq C(K,E)$ ...
0
votes
2answers
40 views

Do metrizable spaces have a countable basis?

Let X be a compact Hausdorff space. If it has a countable basis, is it metrizable? And if it is metrizable, does it have a countable basis?
0
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0answers
20 views

An example of infinite Betti number refers to G-covering, and the motivation of G-covering

I saw from a book that if G is infinite, then when considering G-covering: $p:X'\to X$, the P-th Betti number of $X'$ can be infinite. Can you give me an example of that. Also I am curious why we need ...
0
votes
1answer
16 views

ordered set notation in functions

Do please forgive me, if this question is a duplicate. How does one correctly notate a function $f$, which takes a ordered subset $S$ from the field $\mathbb{K}$ and returns an other (ordered) subset ...
2
votes
3answers
39 views

Homeomorphic but not equivalent compactifications.

I stumbled upon the definition of equivalent compactifications which is: Two compactifications $Z_1$ and $Z_2$ of the space $X$, are said to be equivalent if there exists a homeomorphism ...
0
votes
1answer
24 views

If a linear map $T:X^*\to X^*$ is norm-norm continuous, is it weak-star - weak-star continuous?

Let $X$ be a Banach space and suppose $T:X^*\to X^*$ is a linear mapping. If $T$ is norm-norm continuous, i.e. continuous from the normed space $X^*$ into the normed space $X^*$, is it also continuous ...
0
votes
2answers
25 views

Continuous operators

We have that $T:E\rightarrow \mathbb{R}$ is linear where $E$ is a normed space we have that $\ker T=\{x\in E, Tx=0\}=T^{-1}(\{0\})$ if we suppose that $\ker T$ is closed, as $\{0\}$ is closed can we ...
0
votes
1answer
14 views

Closure of set intersection neighborhood with the closure set

What's the meaning of the definition of the closure set that if (x belongs to A ) the neighborhood of x intersects with A does not empty I'm trying to prove that the closure of A is the smallest ...
1
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0answers
21 views

Example of two affine varieties $X,Y$ such that the image of $\phi:X \rightarrow Y$ is not locally closed

In my course Algebraic Geometry I always find it hard to come up with examples or counterexamples. For instance in the following question: Give an example of two affine varieties $X,Y$ and a morphism ...
1
vote
2answers
54 views

Topology, locally-compact Hausdorff space

I already asked this question here: locally-compact Hausdorff space, equivalent, compact, continuous So if this repost is not apprechiated, please just delete this thread, but I would really like to ...
0
votes
1answer
22 views

Example of a locally compact metric space which is $\sigma$-compact but not proper

Let $(X,d)$ be a locally compact metric space. Then it is known that $X$ is separable if and only if it is $\sigma$-compact (i.e. it can be written as a countable union of compact sets). Moreover, ...
1
vote
1answer
28 views

Is the Kähler differential of a continuous function ring trivial?

Suppose $A=C^0(\mathbb R)$ is the ring of real-valued continuous functions on $\mathbb R$. Is it true that, the Kähler differential $\Omega_{A/\mathbb R}$ trivial? In other words, suppose that ...