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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1answer
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What conditions a function$ f: X \to Y $Must satisfy so that the inverse image of compact is also compact?

If f is closed such that the inverse image of a unitary set is compact, I think it's worth it. But are there other conditions?
17
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2answers
6k views

Nonexistence of an injective $C^1$ map between $\mathbb R^2$ and $\mathbb R$

I am getting bored waiting for the train so I'm thinking whether there can exist a $C^1$ injective map between $\mathbb{R}^2$ and $\mathbb{R}$. It seems to me that the answer is no but I can't find a ...
5
votes
2answers
337 views

Every Topological Vector Space is Regular

Let $V$ be an abstract topological vector space over topological field $K$ (We may assume that $K = \mathbb{C}$ or $K = \mathbb{R}$ for simplicity). This means that the only think that we are allowed ...
0
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1answer
25 views

Cardinality of the family of all possible dense subsets(dense in |R^n) of |R^n

Actually, for each irrational number i, the set {m+ni | m,n are integers} , is dense in |R, as there are continuum many irrationals, hence obviously there are at least continuum many dense sets . And,...
9
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0answers
390 views
+50

Is the graph of the Conway base 13 function connected?

IVT Property: If $a<b$ and $y$ is between $f(a)$ and $f(b)$, then there exists $c\in(a,b)$ such that $f(c)=y$. Theorem. Let $f:\mathbb R \to \mathbb R$ be a function with the IVT Property. If ...
9
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2answers
714 views

“Every open cover admits an open locally finite refinement” - can this refinement always be realized in terms of basis sets?

Let $X$ be a paracompact topological space, let $C$ be an open cover, and let $\mathscr B$ be a basis for the topology. Does there always exist a locally finite refinement that consists of basis sets? ...
0
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0answers
10 views

Density after removing a meagre subset from a non-empty $G_{\delta}$ subset.

Let $X$ be a topological space, $G\subseteq X$ be $G_{\delta}$ and $N\subseteq X$ be meagre in $X$ (and I don't know if this implies that $N$ is meagre in $G$). Question 1. Is $G\setminus N$ dense in ...
3
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1answer
21 views

Does $C_c(X)$ separate points in $X$ when $X$ is a Banach space?

Suppose that $X$ is a separable, infinite dimensional Banach space. We say that a set of functions $\{f_\alpha\}_{\alpha \in A}$ separates points in $X$ if for every $x,y \in X$, there is an $\alpha$...
13
votes
4answers
1k views

Is Top provably not cartesian closed?

It is often said that the category $\sf Top$ of topological spaces and continuous mappings is not cartesian closed. E.g., in the Wikipedia article on compactly generated spaces and in an answer on ...
3
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1answer
50 views

Does classification of 1-manifolds with boundary give induced orientation of image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
9
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1answer
149 views

Simple example of product not preserving coequaliser in $\mathbf{Top}$

In the category of topological spaces ($\mathbf{Top}$), products do not always preserve colimits. If they did then $\mathrm{Hom}_\mathbf{Top}(-\times X,S)$ would be representable and hence $\mathbf{...
4
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0answers
44 views

Elementary example of Baire spaces whose product is not Baire

It is known that there are Baire spaces $X$ and $Y$ whose product is not Baire, the simplest construction I know is due to Cohen and goes as follow: Let $S$ be a stationary subset of $\omega_1$, then ...
5
votes
0answers
55 views

Do products preserve colimits in the category of locales?

Is the functor $X\times-:\mathbf{Loc}\to\mathbf{Loc}$ colimit preserving for all locales $X$? The reason that I'm interested in this question is that the same property fails in the category of ...
1
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1answer
44 views

For local diffeomorphisms,is the $f(U)$ open in range assumption redundant?

My book is An Introduction to Manifolds by Loring W. Tu. From Wikipedia: Local diffeomorphism: For $X$ and $Y$ differentiable manifolds. A function $f:X\to Y$, is a local diffeomorphism, if for ...
31
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4answers
1k views

Can a nowhere continuous function have a connected graph?

After noticing that function $f: \mathbb R\rightarrow \mathbb R $ $$ f(x) = \left\{\begin{array}{l} \sin\frac{1}{x} & \text{for }x\neq 0 \\ 0 &\text{for }x=0 \end{array}\right. $$ has a graph ...
2
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2answers
1k views

Is $GL(n;R)$ closed as a subset of $M_n(R)$?

Let $M_n(R)$ denote the space of all $n×n$ matrices with real entries. The general linear group over real numbers,denoted $GL(n,R)$, is given by $GL(n,R)=${$A∈M_n(R)|det(A)\neq0$}. Is $GL(n,...
1
vote
2answers
309 views

Proving that a set in which any sequence having exactly one accumulation point converges is compact

I found the following question about compactness: A set $S\subseteq \mathbb{R}^n$ has the following property: if a sequence of $S$ has exactly one accumulation point in $S$, then it converges in $...
2
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2answers
83 views

Slanted separating hyperplanes for convex polytopes

EDIT: A previous version of this question was imprecisely formulated—I am grateful to Theo Bendit for providing a counterexample for that version. Let, for some $n\in\mathbb N$, $X\subseteq\mathbb R^...
1
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2answers
20 views

Question about Topological Subspace definition

Let $(S,\tau)$ be a topological space, and let $H\subseteq S$. Using this definition, we can define the topological subspace $(H,τ_H)$ where $τ_H:=\{ U \cap H : U \in τ \}$. Now, having just begun ...
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1answer
39 views

Evaluating Bases in Topology [on hold]

Let $X=R$ Prove whether the following are bases or not: 1) $B_1 = \{(a,b) \subseteq R: a,b \in R\}$ and $B_2 = \{(a,b) \subseteq R: a,b \in Q\}$ 2)$B_1 = \{[a,b) \subseteq R: a,b \in R\}$ and $B_2 = ...
1
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3answers
26 views

If every Component in a compact space is open then the number of components is finite.

Let $X$ a compact set. Prove that if every connected component is open then the number of components is finite. Ok, $X = \bigcup C(x)$ where $C(x)$ is the connected component of $x \in X.$ I know ...
1
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2answers
41 views

Inverse of a projection function

I have got the following question; Let $(X_1, \tau_1), (X_2, \tau_2)$ be topological spaces and $X = X_1 \times X_2$. Equip $X$ with the product topology $\tau$ so that, by definition of $\tau$, the ...
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0answers
26 views

“Show that a function f : ℝ → ℝ is continuous in the ε − δ definition of continuity if and only if, for every x ∈ ℝ and every open set U…” [on hold]

This is a problem from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. ε − δ definition of continuity: "A function f : ℝ → ℝ is continuous if for every x0∈ℝ and ...
13
votes
2answers
245 views

Transitive action of a discrete group on a compact space

Let $G$ be a discrete countable group acting on a compact, Hausdorff space $X$. Assume that the action is transitive. Namely, $G\cdot x=X$, for all $x\in X$. Does it follow that $X$ is finite? I ...
0
votes
1answer
21 views

Comparison between uniform and box topology on $\mathbb{R}^J$

Was reading Munkres' Topology and got stuck in this. He proves that uniform topology is contained in box topology on $\mathbb{R}^J$ but leaves the case that this inclusion is strict when $J$ is ...
1
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0answers
16 views

Deduction of $H^\star(RP^\infty,Z)=Z[x]/(2x),|x|=2$ from $H^\star(RP^\infty, Z_2)=Z_2[y],|y|=1$

This is a statement made in Hatcher, Algebraic Topology Chpt 3, Sec 2. One can compute $H^\star(RP^\infty,Z_2)=Z_2[y]$ with $|y|=1$. Now from cellular cochain complex $C^\star(RP^\infty, Z)\to C^\...
0
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0answers
17 views

Find the limit point of subsets [0,1] , (0,1) , (√ 2,√ 10) , Z and (-∞,0)- [on hold]

Find the limit point of subsets [0,1] , (0,1) , (√ 2,√ 10) , Z and (-∞,0) Just help me with this
0
votes
1answer
17 views

Connected spaces and constant function

Let $Y$ a discrete space. Prove that a space $X$ is connected if only if every $f:X\to Y$ is constant. My incomplete attempt: Ok, If $X$ isn't connected then there are $A, B $ open sets such that $...
2
votes
1answer
45 views

Continuous function and Topology.

Let $X,Y$ topological space and $A \subset X,$ $A$ is dense in $X$. Prove that if $F:X \to Y$ is continuous and constant in $A$, then $F$ is constant in $X.$ I solved it, but I want to show you just ...
0
votes
1answer
42 views

Proving the Non-Emptiness of an Intersection of Neighborhoods in Euclidean k-Space

This is obviously true but I can't figure out how to prove it: Let $r>0$. Let $\varepsilon>0$. Let $k\in\{1,2, \ldots\}$. Let $a\in\mathbb{R}^k$. Let $p\in\mathbb{R}^k$ satisfty $|p-a|=r$ ...
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2answers
33 views

Nonempty boundary of a set?

In the Wikipedia article on the boundary of set, one can read that the boundary of a set $A$ is empty if and only if $A$ is clopen. In case the boundary of a set in not empty, what does it contain?
1
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1answer
17 views

Decomposing a sequence with countable number of accumulation points into convergent sub-sequences in metric space

Let $S$ be any metric space and the sequence $\{x_n\}$ has an infinite but countable number of accumulation points $y_1,y_2,...$ Prove that it is possible to split the index set $\mathbb N$ into a ...
2
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1answer
73 views

Is there any injective parametrization of Klein bottle?

"Let $K$ be (the topological space that is known to topologists as) the Klein bottle. There's a standard immersion $f:K \to \mathbb{R}^3$, whose image is known, in popular culture, as "the Klein ...
0
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0answers
27 views

Topological proof about polygon divided by line in $\mathbb{R}^2$

Let $\mathcal{P}$ be a compact polygon in $\mathbb{R}^2$, and let $\mathcal{R}$ be a line. Show by topological arguments (i.e. using concepts and theorems from general topology) that there exists a ...
4
votes
1answer
35 views

Dimension of a preimage

Suppose we have a differentiable function $f:\mathbb{R}^{k}\to\mathbb{R}^{\ell}$ where $k>\ell$. How can we formalize the fact that the "inverse" of a point $\mathbf{y}\in\mathbb{R}^{\ell}$, $f^{-1}...
8
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2answers
5k views

Show that the boundary of a set equals the boundary of its complement

$\newcommand{\bdy}{\operatorname{bdy}}$ I'm trying to show that $\bdy(A) = \bdy(A^c)$. I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \...
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0answers
61 views

Rudin exercise 2.27

Does the below proof for exercise 2.27 look right? In particular the last part about E containing at most countably many points not in P. Thank you. Q. Suppose $E \subset \mathbb{R}^k$, $E$ is ...
0
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2answers
34 views

$\mathbb{R}^3$ and closed set.

Prove that every plane in $\mathbb{R}^3$ is a closed set. I know a way to solve it, that is, Let $ \pi : ax+by+cz+d=0$ a plane in $\mathbb{R}^3$ and $p=(a,b,c) \in \mathbb{R}^3$ \ $\pi$. Let $0 <...
1
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1answer
41 views

On the proof of $\mu(U)-\mu(K)<\epsilon$ and $\chi_K\le f\le\chi_U$

Let $(X,\tau,\mathcal A,\mu)$ a space that satisfy hypothesis $(H)$. Prove that for all compact $K\subseteq X$ and $\epsilon>0$, exist $f\in C_c(X)$ such that $f(x)=1,\forall x\in K$ and $\mu(K)\le\...
1
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1answer
51 views

Continuity of vector-valued function

Let $X$ and $Z$ be two Banach spaces such that $Z\subset X$ with dense and continuous emmbedding. Let $\phi \in C([0,2],X)$(with respect to the norm of $X$) be a decreasing function such that $range(\...
7
votes
3answers
90 views

Is $T(C) \subseteq R^m$ closed?

Let $C \subset R^n$ be a closed, convex cone. Let $T: R^n \to R^m$ be a linear transformation. Is $T(C) \subseteq R^m$ closed? I'm very positive that the answer is NO. But I couldn't come up to a ...
10
votes
8answers
945 views

Definition of topological space: Is Ω equal to the powerset of X?

A topological space is a set $X$ and a collection $\Omega$ of subsets of $X$ such that: $\emptyset \in \Omega$ and $X \in \Omega$ The union of any collection of $\Omega$ is in $\Omega$ The ...
3
votes
1answer
65 views

Is $C_1 + C_2$ closed?

Two questions jump into my mind when I was working with cone, I feel they are very related to each other. one I asked here "Is $T(C) \subseteq R^m$ closed?" Second one is: If $C_1$ and $C_2$ are two ...
0
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1answer
777 views

How do I calculate the Jacobian matrix of the transformation of a 1-m manifold to a chart (topology question)?

What I want to do is take a 1-m manifold (something like a circle), and transform a subset of that manifold into a chart. I want to represent that function from manifold to chart with a 1 x 1 matrix, ...
7
votes
2answers
48 views

Are countable topological spaces second-countable?

Are countable spaces (i.e. $\mathbb{N}$ with any topology) second-countable? A countable space can have at most $2^\omega$ open subsets which suggests that a counterexample may exist. On the other ...
4
votes
1answer
50 views

Axioms for homology and cohomology for CW complexes

This is related to assertion made in Hatcher, Algebraic Topology, Chpt 3, Sec 1 and Chpt 2. I will write down axioms for cohomology and axioms for homology is written in a similar fashion. ...
0
votes
1answer
39 views

Obtaining mapping cone complex from $f:X\to Y$

This is related to Hatcher, Algebraic Topology Cor 3A.7(b). Cor 3A.7(b) $f:X\to Y$ induces integral homology isomoprhism iff it induces isomorphism with coefficients over $Q$ and for all $Z_p$. "...
1
vote
2answers
228 views

if $X$ is $T_1$ and limit point compact then $X$ is countably compact

A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. I want to show that if $X$ is a $T_1$ space and limit point compact ...
0
votes
1answer
19 views

Connected sets and irrational numbers [duplicate]

Prove that the set $C$ of points in $\mathbb{R}^2$ such that at least a coordinate is irrational is connected set. Ok, I know that the set of Irrational numbers isn't connected but how that helps me? ...