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; ...

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5
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1answer
745 views

existence of a 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 ...
1
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4answers
477 views

Intuitive significance open sets (and software for learning topology?)

I have just started to learn topology and I referred to some books and online lectures. The problem is that they all talk the same things and are missing the same things. I want to know "what is the ...
2
votes
2answers
149 views

Which functions on N extend uniquely to a continuous function on the Stone-Cech Compactification of N?

The question is exactly the title. Is there a good classification of which functions from $\mathbb{N}$ to $\mathbb{N}$ (or, more generally, from $\mathbb{N}^n$ to $\mathbb{N}$)? Also, what is a good ...
3
votes
2answers
161 views

A question dealing with the compactification of the real line

I've been reviewing my old exams, and I came across a question that I was unable to answer. Can anyone help me out on this one? Let $X = \mathbb{R}$, and let $cX$ be the compactification ...
4
votes
1answer
82 views

Grassmanians $Gr_k(\mathbb R^n) \cong Gr_{n-k}(\mathbb R^n)$

I am trying to prove that the Grassmanians $Gr_{n-k}(\mathbb R^n)$ and $Gr_{k}(\mathbb R^n)$ are homeomorphic. Intuitively, this makes sense; specifying a $k$-dimensional subspace is equivalent to ...
1
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2answers
122 views

Which separation axiom?

Let $X$ be a topological space. Assume that for all $x_1,x_2 \in X$ there exist open neighbourhoods $U_i$ of $x_i$ such that $U_1 \cap U_2 = \emptyset$. Such a space, as we all know, is called ...
4
votes
2answers
104 views

A question dealing with Lindelöf spaces

I've been going over previous exams that I've had in topology to study for my comprehensive exam, and I noticed a problem that I missed. I was wondering if anyone could help me out with this problem: ...
1
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4answers
2k views

Showing that $[0,1]$ is compact

Let's choose an open covering for $\left [ 0,1 \right ]$. For example $$\left \{ \left ( \frac 1 n,1-\frac 1 n \right ) \mid n\in \{ 3,4,\dots\} \right \}.$$ How can one choose a finite open ...
1
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2answers
108 views

Does the interior of a Kuratowski 14-set in a finite space always have cardinality 1?

A subset A of a topological space X is called a Kuratowski 14-set if exactly 14 different sets (including A) can be obtained from A by alternately taking closures and complements. Let $c$ denote ...
3
votes
1answer
177 views

compact Hausdorff

A scattered space is a space for which every not empty subset has an isolated point (equivalently for $T_1$ spaces, every not empty closed subset has an isolated point). A compact Hausdorff, not ...
2
votes
1answer
387 views

Use the definition of “topologically conjugate” to prove that $F(x)=4x^3-3x$ is chaotic.

Let $V$ be a set. $f: V \rightarrow V$ is said to be chaotic on $V$ if 1) $f$ has sensitive dependence on initial condition, i.e. there exists $\delta>0$ such that, for any $x\in V$ and any ...
0
votes
2answers
260 views

Disconnectedness of the rationals with the subspace topology

I have tried to prove that $\mathbb{Q}\subset \mathbb{R}$ equipped with the subspace topology is a disconnected space. I'd like to make sure my attempted proof is correct since topological properties ...
0
votes
1answer
180 views

A basis for some topology on $\mathbb{R}^2$

I know that the collection $$\mathcal{B}= \{(x,y): a<x\leq b, c<y\leq d\}$$ is not a basis for the standard topology on $\mathbb{R}^2$, but the collection of open rectangles in the plane ...
1
vote
1answer
384 views

classification of 1-manifolds

i read that the circle $S^1$ is the only connected compact 1-manifold but don't we have that the interval $I=[0,1]$ is a connected compact 1-manifold and that is not homeomorphic to $S^1$? May be they ...
0
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0answers
147 views

Embedding of a finite simplicial complex

In the paper Hardness of embedding simplicial complexes in $\mathbb{R}^{d}$ the abstract states that a finite simplicial complex of dimension $k$ embeds in $\mathbb{R}^{2k}$ while on page $856$ he ...
5
votes
1answer
84 views

Is this a surjection of rings? What am I doing wrong?

Let $Z\newcommand{\df}{:=}\df\newcommand{\C}{\mathbb C}\C$ and $T\df\C^\times$. Then, the coordinate ring of $Z$ is $\C[z]$ and that of $T$ is $\C[t,t^{-1}]$. Consider another copy of $T$ with ...
1
vote
1answer
116 views

Characterising continuous maps between metric spaces

Let $f:(X,d)\to (Y,\rho)$. Prove that $f$ is continuous if and only if $f$ is continuous restricted to all compact subsets of $(X,d)$. I could do the left to right implication but couldn't do the ...
1
vote
1answer
138 views

manifold as simplicial complex

I want to know the topological relation between a manifold and a simplicial complex. I know that a simplicial complex cannot be a manifold since its a union of simplices which are manifolds of ...
6
votes
2answers
365 views

Question about definition of topological manifold

The following definition of topological manifold is given in Lee's Introduction to topological manifolds (2000) on page 33: A topological manifold is a second countable Hausdorff space that is ...
0
votes
1answer
128 views

Dimension of disjoint union of manifolds

While it is clear that a disjoint union of two $d$-manifolds is a $d$-manifold, it is not clear to me if the disjoint union of a $d_1$-manifold and a $d_2$-manifold is still a manifold and if yes ...
2
votes
4answers
719 views

Continuous Deformation Of Punctured Torus

This is problem 11 (b) from the first chapter of "Basic Topology" by M.A. Armstrong. The author hasn't had time to develop many theorems or mathematical machinery, so this problem should be able to ...
10
votes
2answers
390 views

$\{(x,y)\!\in\!\mathbb{B}^n; -\varepsilon\leq-\|x\|^2\!+\!\|y\|^2\leq\varepsilon\}\approx\mathbb{B}^k\!\times\!\mathbb{B}^{n-k}$

The question is motivated by the notion of handle attachment, Morse theory, critical points of index $k$, Morse lemma, sublevel sets, etc. For $0\!\leq\!k\!\leq\!n$ and ...
1
vote
1answer
77 views

Conditions to ensure this union is closed

Let $(M,d)$ be a metric space and let $\{S_n\}_n$ be a countable collection of non-empty closed and bounded subsets of $M$ Are there any additional conditions on the collection$\{S_n\}_n$ to ensure ...
4
votes
1answer
732 views

Proving that the join of a path-connected space with an arbitrary space is simply-connected

I am struggling with the following question from Allen Hatcher's algebraic topology book. Define the join $X*Y$ of two topological spaces $X$ and $Y$ to be the quotient of $X\times Y \times I$ under ...
0
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2answers
92 views

What do I need to know to understand the completion of the field of rational functions of a non-singular projective curve?

So the title gives the jist of my question. Specifically, let $X$ be a non-singular projective curve, $P$ a point on $X$, $v_P$ the discrete valuation associated to the ring $\mathcal{O}_P$. Then I ...
1
vote
2answers
137 views

Strengthening of a Kuratowski's theorem about Connected Sets

It is asserted in this article on "Connective Spaces" that in topological spaces, if $A \subseteq B$ are both connected and $C$ is maximally connected in $B - A$, then $B - C$ is connected (page 4, ...
13
votes
0answers
493 views

Understanding Alexandroff compactification

Is the Alexandroff one-point compactification of a locally compact Hausdorff space ($\mathbf{LCHaus}$) a functor to the category of compact Hausdorff spaces ($\mathbf{CHaus}$)? It seems to me that one ...
6
votes
2answers
253 views

When does the product topology have a countable base?

Could any one tell me how to prove this one? The product topology has a countable base if and only if the topology of each coordinate space has a countable base and all but a countable number ...
6
votes
1answer
111 views

Is this a sufficient condition for a subset of a topological space to be closed?

Let $X$ be a topological space, and let $\{U_i\}$ be an open cover. If $Y$ is subset of $X$ such that $Y\cap U_i$ is closed in $U_i$ (for each $i$), does this imply that $Y$ is closed in $X$?
2
votes
2answers
227 views

Existence of infinite discrete family of open sets in a non compact topological space

How can I prove if a Hausdorff topological space $X$ is not compact, then there exist a countably infinite discrete family of open sets in $X$.
1
vote
1answer
171 views

About the term “continuous monotone map”

In this wiki a monotone map is defined, but in this paper in theorem 1.1 the definition of a monotone function is recalled. The first is concerned with points of the image, but the second is about ...
1
vote
2answers
116 views

is the converse true: in a simply connected domain every harmonic function has its conjugate

The question is. Is the converse true: In a simply connected domain every harmonic function has its conjugate? I am not able to get an example to disprove the statement.
0
votes
4answers
811 views

Boundary points

Can you explain me the definition of a boundary point ? The definition is : Let $A \subset \mathbb{R}^{n}$. A point $x \in \mathbb{R}^{n}$ is called a boundary point of $A$ if every neighborhood ...
4
votes
5answers
246 views

topology - analysis Book

I need some notion about topology(I'm very interested in boundary points, open sets) and few examples of solved exercises about limits of functions($f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$) using ...
10
votes
0answers
438 views

A fiber bundle over Euclidean space is trivial.

What's the easiest way to see this? The only thing I could think to do was try to patch together trivializations. I couldn't find a way to make that work. Thank you! edit: For the record, here's why ...
1
vote
1answer
208 views

another topology multiple choice

Let $S^1 = \{(x, y) \in \Bbb R^2 : x^2 + y^2 = 1 \}.$ Let $D = \{(x, y) \in\Bbb R^2 : x^2 + y^2 \le 1 \}$ and $E = \{(x, y) \in\Bbb R^2 : 2x^2 + 3y^2 \le 1\}$ be also considered as subspaces of $\Bbb ...
7
votes
4answers
5k views

example of neither open nor closed set

I need simple very simple example of set of Real numbers (if there is) that is neither closed nor open. And also a very short and simple explanation why it is neither closed nor open. thank you!!
12
votes
4answers
763 views

What is the smallest cardinality of a Kuratowski 14-set?

A subset $A$ of a topological space $X$ is called a Kuratowski 14-set if exactly 14 different sets (including $A$) can be obtained from $A$ by alternately taking closures and complements. Are there ...
1
vote
1answer
103 views

Open Sets - example

I have the following exercise: Prove that $$A=\{(x,y)\in \mathbb{R}^{2} \mid x >0\}$$ is a open set. I try to solve that exercise with the help of definition, so : To prove that $A$ is ...
3
votes
2answers
160 views

Question about topology definition

I am reading a topology definition: Let $X$ be a set and let $\tau$ be a family of subsets of $X$. Then $\tau$ is called a topology on $X$ if: Both the empty set and $X$ are elements of $\tau$ Any ...
1
vote
0answers
86 views

Neighborhood Retraction of Boundary

Here is the problem: If $M$ is a manifold with boundary, then find a retraction $r:U \rightarrow \partial M$ where $U$ is a neighborhood of $\partial M$. I realize that the collar neighborhood ...
1
vote
0answers
102 views

Is there a fractal origami shape that trades volume for area to always keep a flat surface when expanded?

I'm thinking of something like a 2.5D sierpienski type shape. The idea is to enable an lcd type screen that could unfold to "any" size by unpacking space filling elements packed in 3d to a 2d ...
3
votes
1answer
141 views

Expressing $\mathbb{R}$ as the quotient of a disjoint union of unit intervals

I am trying to complete exercise 3.18(a) in Lee's Introduction to Topological Manifolds. The exercise is as follows: Let $A \subseteq \mathbb{R}$ be the set of integers and let $X$ be the quotient ...
3
votes
2answers
153 views

A homotopy question

I already asked about the interpretation of this problem here. Now I would like to ask about the solution. The problem is Let $A\subseteq X$ be a contractible space. Let $a_0\in A$. Is the ...
8
votes
0answers
146 views

Open map which “almost fixes” the boundary of an open ball

We have a continuous function $f:\bar{B}\to\mathbb{R}^n$, where $\bar{B}=\{x\in\mathbb{R}^n:\|x\|\le 1\}$, such that if $\|x\|=1$ then $\|f(x)-x\|<\epsilon$, for a fixed $\epsilon\in(0,1)$. We have ...
3
votes
1answer
124 views

Circle Acting on Circle/Ball

Is much known about $S^1$-actions on the following simple spaces?: 1) $D^2$ the disk 2) More generally $D^n$ the n-ball 3) $S^1$ the circle In particular, does every $S^1$-action on the disk (or ...
12
votes
1answer
525 views

When is $C_0(X)$ separable?

Recall that a compact Hausdorff space is second countable if and only if the Banach space $C(X)$ of continuous functions on $X$ is separable. I'm looking for a similar criterion for locally compact ...
1
vote
1answer
108 views

Compact subspace

Is the subspace $$C^k([0,T]) \subset C^{k-n}([0,T])$$ compact? I think the answer is no. But since $C^k$ is compactly embedded in $C^{k-n}$, it seems like it should be yes in some way. Can I do ...
0
votes
4answers
280 views

Intersection of compact and discrete subsets

I have difficulties with a rather trivial topological question: A is a discrete subset of $\mathbb{C}$ (complex numbers) and B a compact subset of $\mathbb{C}$. Why is $A \cap B$ finite? I can see ...
3
votes
1answer
327 views

Help proving the primitive roots of unity are dense in the unit circle.

I'm having difficulty understanding how to prove that the primitive roots of unity are in fact dense on the unit circle. I have the following so far: The unit circle can be written ...