Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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26 views

No injective path between the two origins of the “line with two origins”.

During my preparations for my topology exam, I came across the following exercise: Let L denote the line with two origins. Show that there is no injective path between the two origins $0_{-}$ and $0_{...
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17 views

Is $F$ an embedding if instead of $F$ being an immersion we have $\tilde F$ immersion or submersion?

My book is An Introduction to Manifolds by Loring W. Tu. Immersions and submersions are defined here. Smooth embedding is defined here. Is the following correct? Let $N$ and $M$ be smooth ...
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2answers
19 views

How to visualize Dini theorem regarding sequence of functions?

I recently encountered Dini theorem while studying sequence of functions.But the proof is seeming tasteless.I understood the proof that we find in books but I could not interpret it graphically,I want ...
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0answers
16 views

$\tilde F: N \to F(N)$ smooth implies image submanifold

My book is An Introduction to Manifolds by Loring W. Tu. Let Let $N$ and $M$ be smooth manifolds of respective dimensions $n$ and $m$. Let $F:N \to M$ be a smooth map. Let $i: F(N) \to M$ be ...
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1answer
16 views

$\tilde F$ immersion is equivalent to $F$ immersion

Related: Equivalent definitions for smooth embedding?, Are manifold subsets submanifolds? My book is An Introduction to Manifolds by Loring W. Tu. Let $N$ and $M$ be smooth manifolds of respective ...
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11 views

Submanifold implies inclusion is embedding. Why is the inclusion smooth in the first place?

My book is An Introduction to Manifolds by Loring W. Tu. Theorem 11.14 says that regular submanifolds are embedded submanifolds by saying that for a (regular) $n$-submanifold $N$ of an $m$-manifold $...
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35 views

What is the meaning of irreducible manifolds

According to Wikipedia, An irreducible $n$-manifold, is one in which any embedded $(n − 1)$-sphere bounds an embedded $n$-ball. What I understand from this definition is, if $\Bbb S^{n-1}$ is ...
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1answer
10 views

Smoothly homeomorphic for invariance of domain and invariance of dimension

Follow-up to this: Do homeomorphic smooth manifolds, like diffeomorphic ones, have the same dimension? Based on this question Viewing invariance of domain as a converse of invariance of dimension, ...
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1answer
16 views

Prove that if $g$ is injection then $f,g$ are both homeomorphism.

Let $X,Y,Z$ be topological spaces and $f:X \rightarrow Y,g:Y\rightarrow Z$ are both continuos and $g \circ f$ is a homeomorphism. Prove that if $g$ is injection then $f,g$ are both homeomorphism. ...
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3answers
47 views

Question about basis of a topology

The Royden's definition of basis of a topology is exactly as the following. Definition For a topological space $(X, \tau)$ and a point $x$ in $X$, a collection of neighborhoods of $x$, $B_x$, is ...
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1answer
46 views

is zero operator a compact operator?

Is the zero operator defined on any Banach space X is compact ? I think this is trivial, because for any bounded set B in X the image set {0} has compact closure as it is finite. Can we generalize ...
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1answer
10 views

Proof verification: $[0,1]^\omega$ in the uniform topology is not limit point compact.

We wish to find a subset $A\subset[0,1]^\omega$ that is infinite and has no limit point. This will prove that $[0,1]^\omega$ is not limit point compact. My attempt: Let $A$ be the set of binary ...
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1answer
40 views

Every Cover of a Compact Real Interval by Open Intervals Has a Finite Subcover where only Consecutive Sets Overlap?

Intuitively this seems true and would be a useful lemma in proving the fundamental theorem of calculus without assuming continuity of the derivative, i.e. that if $f$ is differentiable on $[a, b$ and ...
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1answer
26 views

Is closed subgroup determined by neighborhood basis?

Question: Let $G$ be a totally disconnected compact Hausdorff topological group, and $V_\alpha$ be a neighborhood basis of the identity. Fixed a closed subgroup $H$ of $G$, and for any $\alpha$ we ...
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1answer
53 views

What is the definition of local diffeomorphism/homeomorphism ONTO IMAGE?

I know what local diffeomorphisms and local homeomorphisms are. I want to know the definition of both local homeomorphism ONTO IMAGE. and local diffeomorphism ONTO IMAGE. I'm CAPITALIZING because of ...
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1answer
22 views

Optimization - Problems with equality constraints [on hold]

To make an open box, we take a rectangular piece of lamina, cut off square pieces from the corners, and bend the remaining lamina into a box shape. Determine the optimal size of the squares, such that ...
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1answer
44 views

Proof verification that bijective local diffeomorphisms are diffeomorphisms

My book is An Introduction to Manifolds by Loring W. Tu. I think that since diffeomorphisms are equivalent to surjective smooth embeddings, I think this is the same as proving an injective local ...
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1answer
32 views

Two kinds of Metrics of Convergence in Measure

There are many kinds of metrics that can induce the topology of convergences in measure. Two most common metrics are Here is the first one. $ d(f,g) := \inf_{\delta > 0} \big(\mu(|f-g|>\...
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0answers
28 views

Viewing invariance of domain as a converse of invariance of dimension

My book is An Introduction to Manifolds by Loring W. Tu. Corollary 8.7 is (smooth) invariance of dimension, and Theorem 22.3 is smooth invariance of domain. I view these as converses and think of ...
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1answer
41 views

For manifolds of the same dimension, are submersions equivalent to immersions?

My book is An Introduction to Manifolds by Loring W. Tu. Immersions and submersions are defined here. Let $A$ and $B$ be manifolds with the same dimension $d$, and let $G: A \to B$ be a smooth map. I ...
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2answers
97 views

Do homeomorphic smooth manifolds, like diffeomorphic ones, have the same dimension? [duplicate]

For smooth manifolds $A$ and $B$ with respective dimensions $a$ and $b$. If $A$ and $B$ are diffeomorphic, then $a=b$. I guess the same is true for homeomorphic topological ($C^0$, I guess) manifolds (...
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1answer
13 views

Is there a way to define “area” of a non-discrete & non-empty open or closed bounded subset of complex plane, topologically?

One of the properties of a non-discrete & non-empty , open or closed bounded subset of a complex plane could be "Area". Is there a way to define "area" of such a set in topological terms ?
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1answer
36 views

Proof that map is closed(in Zariski topology)

"Assume that $X = V((z_1 - 1)z_2 - 1) \hookrightarrow \mathbb{C^2}$ and $f(z_1, z_2) = z_1^2(f : X \rightarrow \mathbb{C})$. Show that f is closed map(in the Zariski topology)." The book that I read ...
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1answer
30 views

Prove that $f$ is $\tau_1 - \tau$ continuous on $\mathbb{R}$

We equip $\mathbb{R}$ with the standard topology $\tau$ and $\tau_1$ is topology generating by the basis $\mathcal{B}_1=\left\lbrace [a,b): a<b \right\rbrace$. The map $f:\mathbb{R} \rightarrow {\...
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1answer
26 views

Let $X=\Bbb R$ endowed with the finite complement topology $T$. Is $X$ compact with respect to $T$?

Let $X=\Bbb R$ endowed with the finite complement topology $T$. Is $X$ compact with respect to $T$? Let $\{ T_a \}$ be an open cover. Let $[a,b]$ with the usual topology and $X$ be $[a,b]$ with the ...
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0answers
13 views

Quasiperiodic continuous factor/semiconjugacy which is not a fibration

Say a flow $\Psi$ on the $n$-torus $\mathbb{T}^n$ is quasiperiodic if it is of the form $\Psi^t(x_1,\ldots,x_n) = (x_1 + \omega_1 t \mod 2\pi, \ldots, x_n + \omega_n t \mod 2\pi)$, where the $\omega_i$...
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1answer
40 views

Automating proofs of the fact that $S \subset \mathbb C$ is open when topological continuity is unavailable

There's a good method for proving a subset of complex plane is open that uses continuity. Right now this method is not formally available to me. The best method is to draw a picture and determine the ...
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1answer
54 views

Algorithms for untying knots?

A typical problem I run into while doing anything involving ropes (e.g. climbing or sailing) is how to eliminate tangles efficiently, preferably without having to feed one end of the rope back through ...
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23 views

Proper Map Equivalent Definitions

Let $f: X\to Y$ be a map between topological spaces. $f$ is called proper when every preimage $f^{-1}(K)$ of every compact $K \subset Y$ is also compact. On wiki's page I found following statements: ...
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9 views

A sparsifying max filter that is not a dilation. Is there a name for this operation?

I implemented a sparsifying max filter that takes a sliding window of length $M$ that runs over a time series of length $L$, keeps the maximum value in that window, and put zeros everywhere else at ...
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0answers
35 views

How is Dugundji's coverage of Algebraic Topology? [on hold]

The latter quarter of Dugundji's Topology starts to cover algebraic flavored topological concepts, the chapters being: Homotopy Maps into Spheres Topology of $\mathbb{R}^n$ Homotopy Type Path Spaces; ...
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1answer
25 views

Alternative way to show that the special orthogonal group is compact

To show that the special orthogonal group ${\rm SO}(n,\mathbb R)$, carrying the subspace topology induced by ${\rm Mat}_{n}(\mathbb R) \cong {\mathbb R}^{n}$, is compact many proofs use the Heine-...
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2answers
28 views

Prove that the set of all isolated points of $A$ is countable

Let $X$ be a second countable topology space. Prove that the set of all isolated points of $A$ is countable ($A$ is an arbitrary subset of X). I tried to relate the base which is countable to set of ...
5
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1answer
37 views

Confusion on locally compact Hausdorff

I am confused on the following theorem. Let X be a space. Then X is locally compact Hausdorff if and only if there exists a space Y satisfying the following conditions: (1) X is a subspace of Y (2) ...
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2answers
50 views

$A$ is a dense set in $X$ iff $A$ is uncountable

We equip an infinite and uncountable set $X$ with the topology $$\mathcal{T}=\left\lbrace U \subset X : U = \emptyset \text{ or } X \setminus U \text{ is countable}\right\rbrace.$$ Prove that $A \...
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1answer
27 views

Limits and properties of the set $B=\{\frac{{(-1)}^nn}{n+1}:n=1,2,3…\}$

I have a problem that asks me investigate these things about the set B. $B=\{\frac{{(-1)}^nn}{n+1}:n=1,2,3...\}$ (a) Find the limit points of B. (b) Is B a closed set? (c) Is B an open set? (d) ...
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0answers
17 views

The boundary $\partial S$ of a compact convex set is simply connected if $\dim\partial S\geq2$

I was wondering if anyone has a reference (e.g., a textbook) for the statement in the title: 'The boundary $\partial S$ of a compact convex set $S\subset\mathbb{R}^n$ is simply connected if $\dim\...
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2answers
64 views

Can I define the compactness using the closure or boundary?

I've seen that I can define a topological space and continuous function with the closure or boundary without the open set. Moreover, there are definitions of the connectedness: If $X=X_1\cup X_2$ and ...
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35 views

Confused on weak/weak-* continuity/compactness

I have only recently been introduced to the notions of weak and weak-* topologies. I know the definition of both as well as an idea of what they represent. It is clear through Banach-Alaoglu and ...
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0answers
59 views

Equivalent definitions for smooth embedding?

Let $N$ and $M$ be smooth manifolds of respective dimensions $n$ and $m$. Let $F:N \to M$ be a smooth map. Please verify my proof of the equivalence of the following 2 definitions. From An ...
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88 views

Question about Invariance of Domain Theorem

Dear fellow mathematicians, As you know, the Invariance of Domain Theorem states the followiing: "Let $f$ be an injective continuous mapping from Euclidean space $\Bbb R^n$ to Euclidean space $\...
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5answers
43 views

Question about terminology on topology.

My professor often says that every metric space is a topological space. But reading the definitions of both terms, it does not make sense to me to state it. That every metric space induces a ...
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1answer
31 views

Relatively compact substes of $\bar{\mathbb{R}}_+^2\setminus \{0\}$

A very silly question, yet I'm not 100% I got the good answer: for some $z \in (0,\infty)$, are the sets $E_{1,z}:=\{(x_1, x_2) \in \bar{\mathbb{R}}_+^2\setminus \{0\}: x_2=+\infty, \, x_1 > z\}$ ...
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2answers
29 views

find $\ \mathbb{Q}^0$( interior of $\mathbb{Q}$) in $\mathbb{R}$ in the following cases

find $\ \mathbb{Q}^0$( interior of $\mathbb{Q}$) in $\mathbb{R}$ in the following cases $1. \mathbb{R}$ equipped with co-finite topology $2. \mathbb{R}$ endowed with the co-countable topology ...
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2answers
34 views

Adapting a solution to solve other similar problems (showing a set is open)

I have looked at various proofs of subsets of complex numbers being open and the solutions are all different and look ad-hoc. I'm trying to find a general pattern that can at least solve most such ...
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1answer
38 views

Is $T$ is a topology on $\mathbb{Z}$ ? yes/no

let $n \in \mathbb{Z}^+$ $T = \{ \emptyset \} \cup \{n\mathbb{Z}\}$ Is $T$ is a topology on $\mathbb{Z}$ ? I thinks yes because here $\emptyset$ and $\{n\mathbb{Z}\}$ are in $T$
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10 views

Can we define Vietoris Topology for $n$ -valued fuzzy topological space $(S,\mathcal{O}_S)$ [on hold]

Here $(S,\mathcal{O}_S)$ is a$n$-valued fuzzy topological space i.e., $\mathcal{O}_S$ is the collection of $n$-fuzzy open sets in $S$. So then is it possible to define Vietoris topology on $S$?. ...
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1answer
44 views

Behaviour of direct limits of topological spaces with respect to preimages

Given a continuous map $p:E\rightarrow B$ where $B$ is given by a colimit of $B_{0}\subseteq B_{1}\subseteq B_{2}\subseteq\dots$. We get the canonical induced map $$ colim_{n\in\mathbb{N}} p^{-1}(...
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0answers
47 views

Is there any Hausdorff, $\sigma$-compact and nowhere locally compact space with every non dense open set not $\sigma$-compact?

I was studying some counterexamples, and found out that, if a space $X$ is Hausdorff and nowhere locally compact, then every continuous function $f : X^* \to \mathbb{R}$ is constant, where $X^* $ is ...
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1answer
24 views

How will Urysohn metrization theorem not hold if there is no countable basis, i.e. all basis are uncountable

From what I read, the Urysohn metrization theorem states that a regular space $X$ with a countable basis (which is a normal space since the base is countable) is metrizable. The proof uses the Urysohn ...