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; ...

learn more… | top users | synonyms (3)

1
vote
0answers
45 views

Show that a Set is dense in $(C[0,1],|| \cdot| |_\infty)$ [duplicate]

I have to show that the set: $$M :=\{f\in C[0,1]:\exists L \gt 0 \: \forall x,y \in [0,1] \space \space \space |f(x)-f(y)| \leq L|x-y| \} $$ is dense in $(C[0,1],|| \cdot| |_\infty)$ Any ideas? ...
2
votes
1answer
41 views

Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in ...
0
votes
2answers
66 views

Is a nonempty intersection of a collection of closed sets closed? [closed]

My intuition is yes, but how would you prove it? Would the nonempty intersection of a collection of closed sets always be closed?
1
vote
0answers
25 views

Homology and triangulation of open surfaces

For example I have an open disk, or an open annulus. How do I triangulate open surfaces to find their (simplicial) Homology? Well, I know that open disk and closed disk are both homotopic to a ...
2
votes
1answer
28 views

Some basic question on pasting map from a square to a Klein bottle and homology

Consider a square $S$ which edges identified as follows Let $K$ be a Klein bottle and $p:S\to K$ be pasting map. Let $X$ be the image of the interior of $S$ under $p$ and let $Y$ be the image of a ...
3
votes
2answers
33 views

Compactness implies closedness in $\mathbb R^n$

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\cl}[1]{\overline{#1}}$ $\newcommand{\e}{\varepsilon}$ I am showing from first principles that compactness in $\mathbb R^n$ implies closed-ness. The ...
1
vote
1answer
32 views

The image of an injective function whose domain is a topological space also a topology

Let $(X, T )$ be a topological space, and let $f : X → Y$ be an injective (but not necessarily surjective) function. QUESTIONS. (1) Is $T_f := \{ f(U) : U ∈ T \}$ necessarily a topology on $Y$ ? ...
0
votes
0answers
34 views

Finding a homotopy function to show that two circles are homotopic on $\mathbb{C}\setminus \{a\}$

If I have two counterclockwise circles in $\mathbb{C}$ that both have some point $a$ in their interior, how can I find a homotopy function to show that one is homotopic to the other?
0
votes
1answer
33 views

$A$ is closed if and only if $\mathbb{C}\setminus A$ is an open

Let $A\subset\mathbb{C}$. Prove that $A$ is closed if and only if $\mathbb{C}\setminus A$ is an open subset of $\mathbb{C}$. Let $A$ be closed. Suppose $\mathbb{C}\setminus A$ is not open. That is, ...
1
vote
1answer
21 views

If and only if condition for a product space to be hausdorff.

Is this true $\forall i \in I \ X_i $ is hausdorff $\iff \prod_{i \in I}X_i $ is hausdorff. I understand that $(\rightarrow )$ is true but don't know if $(\leftarrow)$ is true. If the other ...
1
vote
2answers
50 views

Why do equatons of two variables specify curves in $\mathbb{R}^2$?

I suppose to more formally characterize the question more formally, why are all points of the set $\{ (x,y) \mid F(x,y) = 0 \}$ always boundary points (and I believe also never isolated points) in the ...
0
votes
2answers
31 views

prove that $f(\bigcap^n_i X_i) \subset\bigcap^n_if(X_i) $ [closed]

If $f:A \to B$ and $\{X_i\}^n_i$ is a collection of subsets of $A$, how to prove that $$f\left(\bigcap^n_i X_i\right) \subset\bigcap^n_if(X_i) $$ I have a basic understanding of why it is true, but ...
1
vote
1answer
50 views

Open subsets of $\mathbb{R}^2$

Open subsets of $\mathbb{R}$ can be written as disjoint unions of open intervals- can the same be said in $\mathbb{R^2}$? Open subsets of $\mathbb{R^2}$can be written as disjoint unions of open ...
1
vote
1answer
16 views

Equality between weight and density in metric spaces

I have to prove that in any metric (in generalized version metrizable) space weight of the space is equal to its own density. My job done so far: $$(X,\delta)$$ Is topological space with metric ...
1
vote
0answers
23 views

A question about Widely Connected subsets of Euclidean spaces

Let E be a finite dimensional Euclidean space whose dimension is at least two. Can there exist a subset of E that is both dense in E and Widely Connected?
1
vote
2answers
27 views

Let$ (X; T_{\text{cocountable}})$ be an infinite set, show that it is closed under countable intersections.

Also give an example to show that $\mathcal{T}_{\text{cocountable}}$ need not be closed under arbitrary intersections. I was looking for some feedback on my proof: $X\setminus\bigcap_{\alpha \in I} ...
0
votes
1answer
9 views

How to show X is in the co-finite topology

This might be kind of a silly question but I can't fully grasp why the set X on which the cofinite topology is defined would be contained in the Topology. I know that the closure of all open sets U ...
1
vote
0answers
34 views

Relative Homology (Question about Example 2 in Munkres)

I have no problems for $p=0$ case and for $p\geq 2$ it is quite obvious since $C_p(K,v)=C_p(K),\forall p\geq 2$. Now the tricky is for $p=1$. Since the elements of the kernel now not anly map to $0$ ...
2
votes
1answer
39 views

An introduction to torus and a related fundamental questions on quotient mappings

Let us consider the following mapping on the square [0,1] $\ X $ [0,1] as follow : ($\ x $,0)~($\ x $,1) and (0,$\ x $) ~ (1 ,$\ x $) . The quotient space defined on it is called the two ...
0
votes
0answers
11 views

Continuous maps to $S^n$ without antipodal pairs are homotopic [duplicate]

Let $S^n$ denote the unit sphere in the Euclidean space $\Bbb R^{n+1}$, $X$ a topological space, $f,g:X\to S^n$ are both continuous and there doesn't exist $x\in X$ such that $f(x)=-g(x)$, show ...
2
votes
0answers
17 views

Is $C^\infty$ Urysohn lemma true for infinite dimensional Banach spaces?

$C^\infty$ Urysohn Lemma Let $K$ be a compact subset of $\mathbb{R}^n$ and $U$ be open in $\mathbb{R}^n$ such that $K\subset U$. Then, there exists $f_\in C_c^\infty(\mathbb{R}^n)$ such that ...
1
vote
2answers
61 views

Name of the union of a set with its holes

Given an arbitrary connected and compact set $S$ with holes in it, is there a name for the simply connected set formed by the union of $S$ and its holes? For example, let $S = \{x\in \mathbb{R}^n\ |\ ...
0
votes
1answer
27 views

Difference between homeomorphism and quotient map

I am learning topology and would like to know the difference between Homeomorphism and Quotient maps. I will be grateful if anyone can help me in any way.
0
votes
2answers
25 views

The set of singletons is a subset only on the discrete topology

Let X be a set and let B = { {x} : x ∈ X }. I'm trying to show that the only topology on X that contains B on a subset is the discrete topology.I have been stuck on this for quite a while, I can't ...
2
votes
0answers
23 views

Limit and Isolation points.

I have attempted a question that says to prove that the set of isolated points of a countable complete metric space X forms a dense subset of X. My Attempt It has been shown previously that the ...
1
vote
1answer
35 views

Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology. Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: ...
-1
votes
2answers
41 views

Cardinality of the family of all closed subspaces of a separable Banach space

Is it true that the cardinality of the family of all closed subspaces of a separable Banach space is less than or equal to continuum ? (or, is countably infinite?) Thanks for any answer, comment or ...
0
votes
0answers
16 views

Help finding smooth functions that agree on the boundary, but avoid a critical value.

Basically let $U$ be something like a compact neighborhood of $\mathbb{R}^n$ with smooth boundary $\partial U$ and suppose $f:U \rightarrow \mathbb{R}^n$ is smooth. Now fix $x_0 \in \mathbb{R}^n$ with ...
1
vote
0answers
35 views

Given a CW-structure on a space, when can a subspace be realized as the n-skeleton?

So I've been working with CW-complexes in my algebraic topology classes for about a year now, and while I certainly feel that I understand them pretty well at this point, there have been some ...
0
votes
0answers
13 views

Showing deformation retract : $B^3/T^2$, $R^3/S^1$, $S^2 \wedge S^1$

Here what i want to show is $B^3/T^2$, $R^3/S^1$, $S^2 \wedge S^1$, $i.e$, three spaces are deformation retract to each other. Can you give me some hints or concept(?) geometric way to show this? ...
0
votes
1answer
43 views

Prove the integers in the arithmetic progression topology is not compact

I've been studying for my final exam in a general topology course, and I came upon this problem about compactness that I'm have a really tough time solving. Let $a$ and $b$ be integers, with $b\neq ...
0
votes
1answer
19 views

Are the basic open sets of the Baire space closed?

One way to describe the topology of the Baire space $\mathbb{B} = \omega^\omega$ is that the basic open sets are of the form $N_\eta = \left\{ f \in \omega^\omega \middle |\ \eta \subseteq f ...
2
votes
1answer
40 views

Is this an open covering of $X$ which has no finite subcover?

We have the set $X$ which is the union of the sets \begin{align*} & \{(x,y) \in \mathbb{R}^2 : -2 \leq x \leq 2, -1 < y < 1\} \\ & \{(x,y) \in \mathbb{R}^2 : -1 < x < 1, -2 \leq y ...
0
votes
1answer
31 views

Compactness in a vector space

If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...
0
votes
1answer
21 views

Does every locally constant function has finite image imply compactness?

Let $(X,\mathcal{T})$ be a topological space, $A$ be a subset of $X$. A function $f:A\to X$ is said to be locally constant if for every $x\in A$, there is an open neighborhood of $x$ such that $f$ ...
0
votes
1answer
35 views

Prob. 2, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: Does this metric on $\mathbb{R} \times \mathbb{R}$ induce the dictionary order topology?

Here's Prob. 2, Sec. 20 in the book Topology by James R. Munkres, 2nd edition: Show that $\mathbb{R} \times \mathbb{R}$ in the dictionary order topology is metrizable. This question has ...
0
votes
1answer
14 views

How to Show that Points and Closed Sets Can be Separated by Closed Sets in a T3 (Regular) Space

If $X$ is a regular topological space, given $b \in X$, $A \subset X$ closed, $b\not\in A$, there are open sets $U$ and $V$ such that $\bar{U}\cap\bar{V} = \emptyset$ and $A \subset U, b \in ...
0
votes
1answer
25 views

Accumulation point in real spaces

Sequences in $\mathbb{R}^n$ have a unique limit. Is it true that for any sequence which converges to limit exist there exists no accumulation point a such that $x \neq a$. i.e. does unique limit ...
1
vote
1answer
40 views

$A\times B$ connected component implies $A,B$ connect components

$X,Y$ topological spaces. I want to show that if $C\subseteq X\times Y$ is a connected component, then $C=A\times B$ where $A,B$ are connected components of $X,Y$. What I have so far, is that any ...
0
votes
0answers
16 views

a follow up question regrading the local finiteness axiom in the definition of partitions of unity

This is a follow up question to a previous question about partitions of unity: partitions of unity in the proof of the Meyers-Serrin Theorem Since the answer to the second bullet in the linked ...
0
votes
0answers
26 views

Homeomorphism, compactification

Let $\hat{X}$ be a compactification of a locally compact Hausdorff space $X$. Show, that it exists an unique, continuous function $p_{\hat{X}}:\hat{X}\to X^+$, whose restriction on $X$ is the ...
-1
votes
1answer
32 views

A non-Hausdorff space with unique limit

Can I find a topological space $X$ such that every convergent sequence in $X$ has a unique limit in $X$, but $X$ is not Hausdorff?
2
votes
0answers
36 views

Space of matrices having trace zero

I know the definition of connected space. A connected space is in which there is no components. A space X is said to be disconnected if there exists two disjoint non empty open sets U and V such that ...
-2
votes
2answers
28 views

Connected space with disconnected interior? [closed]

Is there a topological space $X$ where: $X$ is connected, $\partial X$ is connected, but $int(X)$ is not?
2
votes
2answers
41 views

Flawed proof that the closure of a set is closed?

So I'm reading baby Rudin's third edition and on page 35, he shows a proof that the closure of a set is closed. (I'm not questioning the result, but it seems to me the proof has a mistake and I'm ...
0
votes
0answers
28 views

Which topology is assumed here?

I'm doing a problem from a previous exam in topology, but I'm uncertain which topology is given to this set. Here's the problem: Let $Y = I^2\backslash\{(x,y) \in I^2 : x = \frac{1}{2}, y > ...
1
vote
0answers
35 views

Questions about complexes and homology

I just learn about the simplicial and delta complexes and computing homology group. But I have a few questions: Is there any topological space which cannot be given a delta compplex structure? Is ...
1
vote
0answers
16 views

Dual of a point which is in the convex cone of a set, contains the dual cone of that set

Let $\Lambda\subseteq R^n$ contains $m$ elements, where $\lambda_i$ is the $ith$ element, and $co(\Lambda)$ is the smallest convex cone contains $\Lambda$. Also, consider any point $u\in R^n$. Now, I ...
0
votes
1answer
36 views

How is the Cofinite Topology on an infinite set even possible? [duplicate]

To be a topology we need the empty set to be open but- X is not finite so the empty set can't be in it and if X was finite every subset of X would be in the topology...
0
votes
1answer
46 views

Compactification, definite function

Let $\hat{X}$ be the compactification of a Locally compact Hausdorff-space $X$. Show, that it exists an unique, continuous function $p_{\hat{X}}:\hat{X}\to X^+$, whose restriction on $X$ is the ...