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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prove that the family of the cross product of open sets Oi: Oi is in Bi, i = 1, 2…

Prove: Let $B_1, B_2,\ldots, B_n$ be the bases for topology spaces $(X_1,T_1), (X_2,T_2),\ldots,(X_n,T_n)$, respectively. Then the family $$\{U_1 \times U_2 \times\ldots\times U_n: U_i\in B_i, i = ...
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72 views

How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
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84 views

What kinds of structures support integration?

I am doing topology which neatly generalizes analysis, which led me to wonder naturally about generalizations of calculus. Specifically I'm interested in knowing what is required of a mathematical ...
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2answers
56 views

Show that $[a,b] \cup [c,d] $ is not connected

Show that $[a,b] \cup [c,d]=T$, where $a < b$, $c < d$ and $b < c$ is not connected (subset of R) In most books (as Rudin) there are propositions that uses the fact there exists $b < x ...
3
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1answer
77 views

'Compactness' vs 'Closed and bounded' for general metric spaces

We know that for $A \subset \mathbb{R}^n$, A is closed & bounded $\iff$ A is compact and that this does not generalize to general metric spaces. 1.) For which class of metric spaces, is ...
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2answers
32 views

If every subset of X is open then there are no limit points in X

Let $(X,d)$ be a metric space. Then if every subset in $X$ is open there are no limit points in $X$. This is probably a very simple question, but I just can't seem to get anywhere with it.
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2answers
283 views

Give an example of a nonempty finite set which is neither open nor closed?

I had this question that I was struggling to come up with an example for (there may not be an example, in which case why?): Give an example of a nonempty finite set which is neither open nor ...
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3answers
77 views

Topological spaces with prescribed fundamental groups

The question I am about to ask could have gone to the chat section but I want to have the answers/comments in an easy-to-refer-back-to style. For (connected, pointed) topological spaces with trivial ...
2
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1answer
83 views

Is the map from the circular half cone to the $xy$ plane a local isometry?

This is a text book exercise. And I think that this map is not a local isometry. But, I don't know how to show this question. Please help me explaining this question. Thanks a lot. I posted its ...
2
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0answers
80 views

Continuity of functional

This is a follow up question to this previous one. Let $X\subset\mathbb{R}_+$, where $X$ is countable and we are considering the space $\mathbb{R}^X$ of functions $f:X\to\mathbb{R}$ with the topology ...
4
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2answers
137 views

Prove that Baire space $\omega^\omega$ is completely metrizable?

When I tried to prove that Baire space $\omega^\omega$ is completely metrizable, I defined a metric $d$ on $\omega^\omega$ as: If $g,h \in \omega^\omega$ then let $d(g,h)=1/(n+1)$ where $n$ is the ...
6
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1answer
132 views

Maurice Frechet's 1904 Definitions of Compactness

I'm writing a small paper on the history of compactness. Frechet wrote in French, and I don't speak French, so I've been consulting this paper: Taylor, A.E. On page 244, I read that Frechet proved ...
3
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3answers
68 views

Specifying a topology by connected subspaces

Let $\Gamma_1$ and $\Gamma_2$ be topologies on a set $X$ such that: for every subspace $A \subseteq X$, $A$ is connected in $\Gamma_1$ iff it is connected in $\Gamma_2$ Are $\Gamma_1$ and ...
2
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1answer
58 views

Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
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1answer
61 views

The question related to a regular surface.

Prove that an equation of the form $f(x,y,z)=c$ determines a regular surface if $f$, defined on some open subset $S$ of $\mathbb{R}^3$, is smooth and $\nabla f\neq 0$ everywhere in $S$. I know ...
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1answer
117 views

Countable product of complete metric spaces [duplicate]

Someone that can give me a proof of that a countable product of complete metric spaces is complete ?
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0answers
50 views

$C(X,Y)$ complete

I want to prove that: $C(X,Y)$ is complete in the compact-open topology, when every component of $X$ is locally compact with a countable base, and $Y$ is a complete metric space. The proof I am ...
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1answer
25 views

Action of discrete subgroups E(n) on $\Bbb{R}^n$

Isometry group of euclidean space $\Bbb{R}^n$ is displayed by E(n). We say that a subgroup G of E(n) is discrete if and only if the subspace topology (from E(n)) on G is discrete. If X and Y are ...
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1answer
48 views

Question about the proof of $S^3/\mathbb{Z}_2 \cong SO(3)$

I'm trying to show $S^3/\mathbb{Z}_2 \cong SO(3)$ completely rigorously. For that purpose I considered three-sphere $S^3$ as a subspace of the ring of quaternions $\mathbb{H}$ and looked into the map ...
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1answer
30 views

separable space and open covers

If a topological space $X$ is separable, then every open cover of $X$ must be countable? since $X$ is separable , then there exists a countable dense subset $S$. This implies, in every open cover any ...
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1answer
49 views

Convergence functions

Let X be a nonempty set. I define a convergence function on X to be a partial function from the set of all sequences in X, to X, that satisfies the five additional conditions: Every constant ...
0
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1answer
27 views

Exponential of a complex line

Is there an "elementary" way to prove that if $D$ is a one-dimensional vector space in $\mathbb{C}$ (considered here as a real vector space), then $\exp(D) \neq \mathbb{C}^{\ast}$ ?
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1answer
88 views

finding an explanation from munkres about quotient topology

I am trying to learn quotient topology from the book of Munkres where he writes that Let X be the closed Unit ball $\{x\times y : x^2+y^2 \le 1\}$ in $\mathbb{R}^2$, and let $X^*$ be the partition of ...
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1answer
59 views

Orbits of discrete groups

Notice that isometry group of euclidean space$\Bbb{R}^n$ is displayed by E(n). I would like know that why any discrete subgroup G of E(n) ( i.e subspace topology (from E(n)) on G is discrete) has ...
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1answer
54 views

About the continuity of a function in the closed graph theorem proof

I'm reading Functional Analysis book of Rudin, and in the proof of the closed graph theorem, there's one point that I don't understand. Can someone please explain it to me? I really appreciate this. ...
3
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1answer
68 views

Question about topological spaces

Let's mark the standard topological space on $\mathbb{R}$ with $\tau$. We'll define new topology on $\mathbb{R}$, $\tau_l$ with the following base: $B_l = \{[a,b)|a,b \in \mathbb{R},a<b\}$ I have ...
0
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1answer
34 views

Intersection of Dense Sets of Parallel Lines

The question is: In $\mathbb{R}^2$, with the usual topology, let $S_i$ be a dense set of points that can be partitioned into parallel lines. Let $m_i$ be the slope of these lines. For any $n\in ...
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1answer
67 views

Compactness and the suspension of a topological space

I would like to prove the following statement: A topological space $X$ is compact if and only if its suspension $SX$ is compact. The proof in one direction is pretty easy: If $X$ is compact, then ...
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2answers
35 views

Meaning of Quotient in this context

I was seeing the following problem a couple days ago: Let $R \subset \mathbb{R}^2$ denote the unit square $R = [0,1] \times [0,1]$. If $F \subset R$ is finite, is $R \backslash F$ connected? I ...
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1answer
241 views

Minimum of two metrics is metric

Let X be a set and d, e two metrics on it. How can I show that the function f defined by f(x,y) = min(d(x,y), e(x,y)) is again a metric? This has been bothering me for the past few days as it seems ...
2
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1answer
98 views

Even trigonometric functions are dense

Let $C([0,K])$ be the space of continuous functions on $[0,K]$ for $K>0$. Consider all linear combinations of $(\cos(a x))_{a\in \mathbb{N}}$ defined on $[0,K]$. Is it true that these are dense in ...
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1answer
47 views

Normal Space New Properties

Assume $X$ is $T_1$-space. prove that, $X$ is normal if and only if for every closed set $C\subset X$ and open set $U\subset X$ such that $C\subset U$ there exist an open set $V\subset X$ such that ...
0
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1answer
36 views

Coincident boundaries of sets

At the end of my lecture, which introduced closed sets and boundaries of sets, my professor asked the following question: Let $S,T,V \subset \mathbb{R}^2$ such that $int(S) \not = \emptyset$, $int(T) ...
0
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1answer
31 views

K-Topology on the real line

The K-topology on the real line by taking as basis all open intervals $(a,b)$ and $(a,b)\setminus B$ where $K=\{1/n:n=1,2,...\}$ and $B\subset K$. According to this topology,the subset ...
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2answers
33 views

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$ Where $\Omega = \{(x,y,z) : z \leq 1\}$ and $\rm{Fr}(\Omega)$ means the $\Omega$'s boundary.
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1answer
39 views

Question about the continuity property of a function in topological vector space

I'm reading Functional Analysis book of Walter Rudin, and there's one point in this book that I don't know why he states that. Here is the statement: $f$ is a linear mapping from F-space $X$ into ...
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0answers
27 views

Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
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0answers
95 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
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1answer
56 views

An alternative definition of locally finite family in $\mathbb{R}^n$

A family $\mathcal{F}$ from subsets of $\mathbb{R}^n$ is said to be locally finite if for each point $x\in \mathbb{R}^n$ there is a neighborhood $U$ of $x$ such that $U\cap F\not=\phi$ only for ...
0
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1answer
39 views

A function extension problem

Let $f: A \to Y $ be a continuous function, where $A$ is a closed subset of a space $X$, then is it true that $f$ can always extend to a continuous function $U \to Y$ for some open neighborhood of ...
6
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3answers
173 views

How much topology for graph theory?

I am writing a thesis in the context of descriptive complexity in theoretical computer science and therefore need to study a little bit of graph theory. My background is not mathematics but computer ...
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3answers
141 views

Quotient space is connected…

Let $X$ be a topological space & ~ be an equivalence relation defined on it.Let $Y$ be the space $X/$~ & $p :X \rightarrow Y $ be the quotient map, $Y$ is given the quotient topology.Given Y ...
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39 views

Example of a continuous bijective function on and to the closure of the complex numbers, with an inverse that is not continuous?

Note that by the closure of the complex number, I mean the union of the complex numbers and infinity. I have been stumbling over this questions for a wile now, and I understand many examples of this ...
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1answer
54 views

The Tangent Disc Topology

Let $X$ be the tangent disc topology, $X=P\cup L$ where $P=\{(x,y):x,y\in \mathbb{R}, y>0\}$ and $L$ is the real line.Then, $X$ is completely regular but not normal, $X$ is separable, $X$ is ...
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3answers
150 views

What is an awesome book as an introduction to hyper groups

I'm a grad studen and i'm choosing an area to follow on my doctorate (in?) and I've been thinking about extension of topological group theory results to topological hypergroups, but for that i need to ...
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0answers
355 views

Is the boundary of a connected set connected?

SOLVED - check out the comments or directly this answer. I was wondering if the boundary of an unbounded component $C$ of the complement of a bounded connected open set $U$ must be connected (in ...
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1answer
436 views

Must the intersection of connected sets be connected?

Must the intersection of two connected sets be connected? I believe the answer is no, but I am not entirely sure. I think a counter example would be a set that intersects another set in more than one ...
3
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1answer
48 views

Countability Axioms and $T_0$ and $T_1$ space

Consider $\mathbb{N}$ as natural number with following topology $T$ : $U\subset \mathbb{N}$ is nonempty, $U\in T$ if and only if $U$ has the property that natural number $n$ belongs to $U$ only if ...
3
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2answers
36 views

Is every regular paratopological group completely regular?

This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open?
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1answer
57 views

Cauchy sequence with respect to Hausdorff metric

We know that if $(X,d)$ is a complete metric space, then $(CB(X),H)$ is complete too, where $CB(X)$ is the collection of non-empty closed bounded subset of $X$ and $H$ is the Hausdorff metric induced ...