**32.803 general-topology questions.**

Let $A$ be an open set according to $\vert\vert \cdot \vert\vert^{A}$.
Prove that $A$ is open according to a different norm $\vert\vert \cdot \vert\vert^{B}.$
Idea: Since $A$ is open, it follows ...

I am looking for a subset $A$ of $\mathbb{R}$ that has the following properties:
It's bounded
It's closed
It's boundary isn't of Lebesgue measure $0$
Edit: My problem was that i just couldn't think ...

Let $Y$ be a linear subspace of codimension 1 in an infinite dimensional Banach space $X$ i.e. $\dim (X/Y)=1$. Then how to prove that $X\setminus Y$ is path connected if and only if $Y$ is dense in $X$...

$A\subset \Bbb{R}$ and let $X(A)\subset \Bbb{R}^2$ be a union of all segments connecting point $(0,1)$ with points $(a,0)$ such that $a\in A$. Show that $X(A)$ with euclidean metric is complete if and ...

Let
$(M,d)$ be a metric space
$E$ be a normed $\mathbb R$-vector space
Now, let $$\left\|f\right\|_{\tilde C^{0+\alpha}(A,\:E)}:=\sup_{\stackrel{x,\:y,\:x',\:y'\:\in\:A}{x\:\ne\:x',\:y\:\ne\:y'}}\...

I want to compare three definitions for an action $G\times X\to X$ to be proper:
For each $x\in X$, there is a neighborhood $U\subseteq X$ such that $gU\cap U=\emptyset$ for all but finitely many $g\...

Let $f : X \rightarrow f(X)$ be continuous and X connected. Then $f(X)$ is connected.
I'm coming across a major problem with the following proof:
Suppose $f(X)$ is not connected. Then $\exists$ non-...

I've got some doubts about the hypothesis concerning the existence of a universal cover for a space.
More precisely, from Hatcher's book I know that:
Every path connected, locally path connected, ...

I'm doing some excercise in Lee's book. I've come up with a solution but not really sure. I also want to see other people solutions too. Here is the problem:
Let $X$ be any topological space and $...

Let $X$ be a non-emptyset and $\mathcal{S}\subseteq 2^X.$ Show that the family
$$\tau(\mathcal{S}):=\{\cup\mathcal{B}^*\mid\mathcal{B}^*\subseteq\mathcal{B}=\{\cap\mathcal{S}^*\mid(\mathcal{S}^*\...

The classification theorem for closed surfaces states that a orientable closed surface is homeomorphic to a sphere with m handles with m >= 0; while a nonorintable closed surface is homeomorphic to a ...

In a topology course we proved the following proposition:
Let $A$ be an infinite set. Then there exists a Hausdorff space $X$ of cardinality $|\mathfrak{P}(\mathfrak{P}(A))|$ which contains a dense ...

This question is related to An example of a compact topological space which is not the continuous image of a compact Hausdorff space?.
Notation: A quasicompact space is one such that each open cover ...

Knowing that $Y\subset X$ is a retract of a topological space $X$ if there exists a continuous function $r:X\to Y$ such that $r(y)=y\quad\forall y\in Y$, I don't know how to show that $\{0,1\}$ is not ...

I have to determine if the next statement is true or false (in the context of differentiable dynamical systems)
A homeomorphism with a backward dense orbit has a forward dense orbit
I believe its ...

In Atiyah and Macdonald chapter 1, it is an exercise to prove the following:
Let $X$ be a compact Hausdorff space. Let $C(X)$ be the commutative ring of $\mathbb{R}$-valued continuous functions on $...

I'm looking for a proof of the following result. I don't know if it is true or not, but it seems to be true. I have tried it from different points of view, but I can not formalize them correctly. My ...

Let $X$ and $Y$ be topological spaces, and $f:X\times Y\rightarrow \mathbb{R}$ be continuous.
Now lets define the function $g:Y\rightarrow X$ as follows:
$$g(y)=\underset{x}{\operatorname{argmin}}f(x,...

$X=\{1,2,3\}$
$\mathcal{T}=\{\emptyset, \{1\}, \{1,2\}, X\}$
How can I show $\mathcal V \subseteq \mathcal T$ is a subbasis of $\mathcal T \Leftrightarrow \mathcal V \supseteq \{\{1\},\{1,2\}\}$?
$\...

Let
$X$ be a normed $\mathbb R$-vector space
$d$ denote the metric induced by $\left\|\;\cdot\;\right\|_X$
$h\in X$ with $\left\|x\right\|_X=1$
$\Lambda\subseteq X$ be open
$E$ be a $\mathbb R$-...

I have this question that says find the homotopy fiber of $S^1⋁S^1↪S^1×S^1$.
can somebody explain the concepts or what needs to be done here?

Can the left adjoint to the inclusion functor $i : \mathbf{Haus} \to \mathbf{Top}$ be constructed (1) constructively, (2) predicatively and (3) in ZF?
If all three conditions (i.e., (1), (2) and (3)) ...

Let $M$ be a dense subset of $\mathbb{R}$.
I want to show that $\{(-\infty, q): q\in M\} \cup \{(q,\infty): q\in M)\}$ is a subbasis of the Euclidean topology.
What's the 'best' way to show this? I ...

Let $\ell^\infty$ denote the space of all bounded real sequences with the usual norm and let $A=\{0,1\}^\mathbb{N}$ denote the set of sequences taking values in $\{0,1\}$. It's easy to see that $A$ is ...

$X$ is a topological space, I need to give example such that for $x_1,x_0 \in X$
$\pi_1(X,x_0) \not \cong \pi_1(X,x_1)$
I think the example is somehow related to the fact that $X$ is not path ...

I am currently reading Brezi's Functional Analysis, Sobolev spaces and Partial Differential Equations. I am somehow stuck in the proof of Theorem 9.2 where the setup is the following:
$\Omega \...

What are the continuous linear functionals on $\mathbb{R}^\mathbb{N}$, when equipped with the product topology? In particular, do they depend on only finitely many coordinates? If not, for what (...

Let $f$ be a continuous mapping of a Hausdorff non-separable space $(X,\tau)$ onto itself. Prove that there exists a proper non-empty closed subset $A$ of $X$ such that $f(A) = A$.
[ Hint: Let $...

Let $H$ be an infinite dimensional Hilbert space and let $e_{n}$ be an orthonormal basis of $H$. Let $\phi$ be a linear functional defined on $B(H)$ as follows. $ϕ(A)=\underset{n}∑(\frac{1}{2})^n⟨Ae_{...

I'm working on a project, which shows the differences between the original Mobius strip, A Mobius strip with an additional even number of half-twists, and A Mobius strip with an additional odd number ...

I just wanted someone to check if my reasoning is all right. I had two approaches to it.
1.
Let $(X,d)$ be a metric space, take any two points $x$ and $y$ in $X$ let $d(x,y)=4\epsilon$ for some $\...

Let's start by looking at the following example from ring theory:
Let $A$ and $B$ be commutative rings with identity and let $f:A\to B$ be a ring homomorphism. For every ideal $I$ of $A$, the ...

I'm trying to do the following exercise
1) Show that $\mathbb{C} \setminus \ (\{1/n \mid n \in \mathbb{N}\} \cup \{0\})$, $\mathbb{C}\setminus \mathbb{N}$ and $\mathbb{C}\setminus \mathbb{Z}$ are ...

If $E \subset \mathbb{R}^n$ and $x \in E$, we have that $ \operatorname{dist}(x,\partial E) = \sup \{ \gamma \geq 0 : B(x, \gamma) \subset E \}$. What characteristics of $\mathbb{R}^n$ make this true? ...

Previously, someone posted his/her proof of "every neighborhood is open" here. I am focusing on the same statement, but it seems that my approach is different from his/hers. Could someone comment on ...

The exercise asks to prove that any pair of three sets $$A=\Bbb{N}\\A'=\{0\}\cup\{\frac{1}{n} | n\in \Bbb{N}\}\\
A''=\{0\}\cup\{\frac{1}{n} | n\in \Bbb{N}\} \cup \Bbb{N}$$ is not homeomorphic.
I ...

I am reading topology by Gf Simmons and I came across a theorem on compactness which stated A topological space is compact if every open basic cover has a finite subcover.
But it defined a basic open ...

S is an Hausdorff topological space. A decreasing nested sequence of non-empty compact subsets of S has a non-empty intersection. In other words, supposing $C_{k}$ is a sequence of non-empty, compact ...

Let $p:(\tilde{X},\tilde{x})\to (X,x)$ be a path-connected and locally path-connected covering space, where $X$ is path-connected, locally path-connected and semilocaly simply-connected. Consider the ...

This is a fairly general question but I couldn't find a source that deals with this problem in the sense I was wondering.
Suppose that we are given two topological spaces $X, Y$ such that $X \times Y ...

I came across this very interesting question, which seems to be partially answered in a couple posts around here:
Let $f:[0,1]\rightarrow\mathbb{R}$ continuous such that $f(0)=f(1)$. Then for all $n&...

I am working with the Volterra equation:
$$b(x)= \int_0^x b(y) k(x,y) dy + f(x) $$
with $b,f \in L^1(\mathbb R)$ and $k \in L^1(\mathbb R^2 )$.
Define $K(b)(x) =\int_0^x b(y) k(x,y) dy $.
I ...

Consider $\mathbb{R}^3/E_i$ with $E_1 = \{(x,y,z) \in \mathbb{R}^2 \vert x^2 + y^2 + z^2 \leq 1\}$ and $E_2 = \{(x,y,z) \in \mathbb{R}^2 \vert x^2 + y^2 + z^2 < 1\}$. I want to study the quotient ...

Basic properties of Thom spaces for vector bundles are rather scattered throughout various sources, mostly without proof: Is it right that
$T(\xi\oplus\zeta)=T\xi\wedge T\zeta$ for two bundles $\xi$ ...

I need to prove the following statement: Let $(E,d)$ be a metric space and let $A, B$ be two disjoint dense subsets of $E.$ Then $\mathring A=\mathring B=\emptyset.$
By definition, $\overline A=E=\...

An answer to the question would be nice, but what I want most is a list of books or principles to help resolve questions like this myself.

How to prove that a weakly normal Banach space is Lindelöf?
It's known that for a Banach space X, the following conditions are equivalent.
(i) X is weakly Lindelöf.
(ii) X is weakly paracompact.
(...

The following proof was suggested: suppose [0,1] was the disjoint countable union of closed intervals. Write the intervals as $[a_n,b_n]$. Start by showing the set of endpoints $a_n, b_n$ is closed. ...

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?

Let $(M,d)$ be a metric space, let $A \subset B \subset M$, and let $x \in A$. Does the inequality below hold?
$$
dist(x, \partial A) \leq dist(x, \partial B)
$$
Follow-up question: if not, does it ...

- real-analysis
- metric-spaces
- algebraic-topology
- compactness
- functional-analysis
- analysis
- continuity
- connectedness
- proof-verification
- elementary-set-theory
- manifolds
- measure-theory
- differential-geometry
- differential-topology
- examples-counterexamples
- homotopy-theory
- geometry
- topological-groups
- reference-request
- calculus
- complex-analysis
- abstract-algebra
- algebraic-geometry
- linear-algebra
- quotient-spaces

- Using Graphics3D to plot sphere upon XYZ coordinate frame
- How do we recover units of force from units of gravitational potential?
- Strange footprint labeled PJ## on laptop's motherboard
- Dealing with a PhD student reneging on an agreement to appear in social media
- Logo design - Finding a good mix between the metaphor and legibility
- Why was the Council of Nicea held in Nicea rather than Rome or Constantinople?
- What did Thanos mean by this?
- If a pilot ejects, what is the autopilot programmed to do?
- What does machine code actually look like while being run?
- Number theory and combinatorial problem
- I am hated by the world
- Why does the Party allow playing chess in Orwell's 1984?
- Is exploratory data analysis important when doing purely predictive modeling?
- Why doesn't SSH use TLS?
- Just the minus sign in the exponent
- What are these Russian characters that aren't in the alphabet in my learning material?
- How fast can I flood the Netherlands entirely and permanently?
- Can I shut off my keyboard w/o turning off my computer?
- Can a planet harbor plants of different colors without one pigment outcompeting the others?
- How to maintain friendship with guys after they're married?
- Let's design a digit mosaic
- Why does Tony Stark say this sentence about foul language?
- Bigger space after itemize/enumerate
- Cambridge Tripos 2015: Zorn's Lemma Proof Requiring Axiom of Choice