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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6 views

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

Let $X$ be an infinite and uncountable set. $\mathcal{T}=\left\lbrace U \subset X : U = \emptyset \text{ or } X \setminus U \text{ is countable}\right\rbrace$. Prove that $A \subset X$ is dense in $\...
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1answer
7 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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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
36 views

Is there a way to describe the compactness as the closure or the boundary in a topological space?

I've seen that I can define a topological space and a continuous function with the closure or the boundary without the open set. Moreover, there are the following definitions of connectedness: If $X=...
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17 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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10 views

Are manifold subsets of manifolds regular/embedded submanifolds?

My book is An Introduction to Manifolds by Loring W. Tu. In groups, vector spaces or rings, if $A \subseteq B$ and they are both rings, both vector spaces or both groups, I think $A$ is not ...
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7 views

Prove embedding (immersion and homeomorphism onto image) is equivalent to image is smooth submanifold and diffeomorphism onto image

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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68 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
40 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
30 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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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
30 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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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
37 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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1answer
43 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 ...
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0answers
36 views

Families of (not necessarily disjoint) topological spaces [on hold]

Is there any theory of families of topological spaces? I do not mean products or disjoint unions. And I do not mean many topologies on one space. And probably I do not speak about subspaces and ...
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1answer
51 views

On Lyapunov numbers

I'm working on a paper , I almost understood the whole paper , I just don't understood some examples of that. I try to explain a little about some definitions. $(X,f)$ is a dynamical system , ...
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1answer
21 views

How to show that a regular space with non-countable basis is not normal

I read that every regular space $X$ with a countable basis is normal. For if $A$ and $B$ are disjoint closed sets in $X$, one can form a neighborhood $U=\bigcup U_j$ over $A$ (from countable bases $...
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1answer
53 views

Approximation of irrationals in $[0,1]$

Enumerate the rationals in $[0,1]$ as follows: $\{0,1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5,... \}$, and let $F_n$ be the set consisting of the first $n$ elements of this enumeration. Is there a sequence ...
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3answers
251 views

Is an infinite set with no limit point unbounded in an arbitrary metric space?

Given an infinite set $X$ with no limit points, is $X$ unbounded? (In an arbitrary metric space) I only know how to do this in $\mathbb{R}^k$. Since $X$ has no limit points, $X$ is closed. An ...
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2answers
41 views

Rudin RCA 5.6-7 Forms of Baire’s Theorem

In Rudin's Real and Complex Analysis, he mentions two (equivalent) forms of Baire's theorem, $(1)$: for a complete metric space $X$, the intersection of every countable collection of dense open ...
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1answer
64 views

Example of a topological space

I have realized that inserting finiteness in topological spaces can lead to some bizarre behavior. For example, it seemed natural to say that every compact subspace of a metric space is closed and ...
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1answer
49 views

Hausdorff spaces from filters

I'm sure I'm just being silly, but I've run into a claim in a paper I'm reading which I don't understand. Suppose $\mathcal{F}$ is a filter on $\mathbb{N}$. There is a natural topology $\tau_\mathcal{...
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1answer
24 views

Convergence in topological space

I need to show that a convergent sequence in a metric space $(X,d)$ converges in a topological space $(X,t)$ where $t$ is the topology generated by $d$ on $E$. By definition, for every $\epsilon >...
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3answers
34 views

Prove that $\mathcal{T}_1 \subset \mathcal{T}$ with $\mathcal{T}_1$ is the standard topology on $\mathbb{R}$.

Let $\mathcal{B}= \left\lbrace [a,b): a,b \in \mathbb{R}, a<b \right\rbrace$. Let $\mathcal{T}_1$ be the standard topology on $\mathbb{R}$ I have already proved $\mathcal{B}$ is a base of a ...
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3answers
55 views

Is $\mathcal{T}$ a topology on $\mathbb{R}$?

Let $a \in \mathbb{R}$, we have $V_a=(a,+\infty)$, $F_a=[a,+\infty)$. 1.Prove that $\mathcal{T}=\left\lbrace \emptyset,\mathbb{R},V_a:a\in\mathbb{R}\right\rbrace$ is a topology over $\mathbb{R}$. 2....
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2answers
77 views

$[0,1]$ is commonly called the unit interval - is there a similar term for $[-1,1]$?

The interval $[0, 1]$ is commonly called the 'unit interval'. Is there something similar for $[-1, 1]$? Like a pre-defined name.
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6answers
226 views

Closure in a topological space

Let $(X, \tau)$ be a topological space. Let $A,B$ be subsets of $X$. Show that $cl(A \cup B)$ $\subset$ $cl(A)$ $\cup$ $cl(B)$ Proof: Let x be in the closure of $A \cup B$. That means for every open ...
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1answer
29 views

Interior topology

Let $(X,\tau)$ be a topological space. Show that $int(A) \cap int(B)$ $\subset$ $int(A \cap B)$ Proof: x $\in$ $int(A)$ and $x\in int(B)$ means that $\exists$ $U_1 (open)$ containing x so that $U_1 ...
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1answer
17 views

Test if line segment formed by two points intersect with image edges

I have a map that looks roughly like the following picture. The map is discretized into pixels and has size nxn where n is in <...
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1answer
46 views

Nagata-Smirnov vs. Urysohn metrization theorems - an example?

I'm looking for an example to demonstrate that fact that the Nagata-Smirnov thm is "more useful" than Urysohn's. That is, I'm looking for a space that you can prove is metrizable using Nagata-Smirnov ...
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0answers
49 views

Given subset $A$ and $ A_x = \{y : (x,y) \in A\} $, Are the following equivalent

(This is Baire-Fubini theorem for categories) For a subset $A \subset \mathbb R^2$ and $x \in \mathbb R$ we denote $$ A_x = \{y : (x,y) \in A\} $$ Are the following equivalent? $A$ is of first ...
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1answer
81 views

Fundamental group of $S^3$ with finitely many points removed?

I had a question about the fundamental group if the $3$-sphere with finitely many points removed. Since $\pi_1(\mathbb{S}^{n-1}) \cong \pi_1({\mathbb{R}}^n\setminus\{0\})$, I thought $\pi_1(\mathbb{S}...
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1answer
18 views

Infinite intersection of nested finite open cover of a compact space

In my research, $X$ is a topological space and for every $n\in\mathbb{N}$, $\mathcal{U}_n$ is a finite open cover of $X$ such that for every $n\in\mathbb{N}$, we have $\mathcal{U}_{n+1}\prec \mathcal{...
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1answer
38 views

Is a compact metric space with a “doubled” point sequentially compact?

Let $X$ be a compact metric space with topology $\tau$ generated by the metric. Consider a new point $x_1\notin X$ and a non-isolated point $x_0\in X$. Set $\overline{X}=X\cup \{x_1\}$, equipped with ...
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1answer
26 views

The Fundamental group of the circle from “Introduction to knot theory”, Ralph H. Fox (3)

In this book,as a third completion (the second completion I have not asked it yet as I am still writing it) to The Fundamental group of the circle from "Introduction to knot theory", Ralph H....
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2answers
67 views

Proving that $(0,1)^3$ not homeomorphic to $[0,3)^3$

What are some of the various ways of proving that $(0,1)^3$ is not homeomorphic to $[0,3)^3$ using the fundamental group and homology groups? I feel like I have various ways of understanding why this ...
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0answers
32 views

Is any space with this property homeomorphic to the three-torus? [on hold]

If you have a three-dimensional, locally Euclidean space such that any path along a coordinate direction is closed (you always eventually come back to your starting point), is this necessarily ...
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1answer
51 views

How do we prove that this set is not manifold? [on hold]

Prove that: $$Z:=\Big[ {(x,y,z) in $R^3$|x^2 +y^2-z^2=0}\Big]$$ is not a manifold.(Even It is not a topology manifold )
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0answers
23 views

Look for a cochain map $ \psi : C_* \to D_* $ [duplicate]

I am looking for a injective cochain map $ \psi : C_* \to D_* $ such as the map {$\psi_i: C_i \to D_i$} is an injective but the map {$\psi_*: H_I(C_*) \to H_i(D_*)$} is not an injective for any $i\geq ...
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2answers
41 views

Topology Hausdorff space

Let $X$ be Hausdorff space and $f$ is a continuous function from $[0,1]$ to $X$. If $f$ is one-one, then image of $f$ is homeomorphic to $[0,1].$ I did something like defining mapping $g$ from image ...
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1answer
18 views

Definition of multi resolution analysis

How can (a) $V_j \subseteq V_{j+1}$ be true, yet (c) is also true? Does that mean that $V_j$ as $j \rightarrow -\infty = \{0\}$?
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1answer
41 views

Order topology on $\mathbb{N}$ is discret topology?

Let $\mathbb{N}=\{0,1,2,\dotso\}$ and $(\mathbb{N},<)$ with the usual ordering $<$. Let $\tau_<$ be the order topology with regards to $<$. Then $\tau_<$ is the discrete topology (...
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3answers
71 views

Every interval $I \subset \mathbb{R}$ is connected. [Proof clarification]

I struggled to understand a part of the following proof. Topological Proof that every Interval $I \subset \mathbb{R}$ is connected Definition: A topological space is connected if, and only if, it ...
2
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1answer
34 views

Determine $X$ up to homeomorphism.

Let $X$ be a non-empty topological space. Assume that every function $f:X \rightarrow \mathbb{R}$ is continuous. Determine $X$ up to homeomorphism, assuming that $X$ is countable. My try: If $X$ is ...
3
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1answer
28 views

$C^m\to C^{m-1}$ projection map projecting out last factor inducing $G_n(C^{m-1})\to G_n(C^m)$?

Assume $m>n$. Consider projection map $\pi: C^m\to C^{m-1}$ by $(z_1,\dots, z_m)\to (z_1,\dots, z_{m-1})$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$ and endow the topology as a ...
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1answer
26 views

Let $h,h'$ be hermitian metric over vector space $V$, then grassmanian $G_n(V_h)\to G_n(V_{h'})$ is always continuous?

Let $V$ be a finite dimensional vector spaces over complex number and choose hermitian metrics $h,h'$ over $V$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$. Since $V$ has 2 metrics, ...
3
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2answers
62 views

Neighbourhoods in topology

Let $X=C[0,1]$ and consider the topology $\tau=\tau(S)$ generated by $$S=\{V_{x,U}\}_{x\in[0,1],~U=(a,b)\subset\Bbb R},$$ where $$V_{x,U}=\{f\in C[0,1]:f(x)\in U\}$$ $1)$ Let $V\in\tau$ be a ...