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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1answer
28 views

$f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function.

Let $U$ be an open connected subset of $\Bbb R^n$ and $f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function. I am facing ...
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1answer
36 views

Characterisation of points of closure in terms of filter bases

I am trying to prove the following: If $E \subset X$, then $x \in \overline{E}$ iff there is a filter base in $E$ converging to $x$. I can only prove the converse implication, as follows. (Any ...
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0answers
31 views

Given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected.

Let $X$ be a (metric) space such that given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected. Let us consider a continuous function $f : X \to ...
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2answers
135 views

Uniqueness of a continuous extension of a function into a Hausdorff space

Suppose that $A\subset X$ and suppose that $f : A \to Y$ is a continuous function with $Y$ Hausdorff. Show that there is at most one continuous function $g : \bar{A} \to Y$. My try: Suppose ...
3
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1answer
230 views

Properties if the one-point compactification of an uncountable discrete space

Let $D ( \tau )$ be an uncountable discrete space, and $\alpha D ( \tau )=D ( \tau )\cup\{\alpha\}$ the one-point compactification of $D ( \tau )$. I want to show that if $U$ is any countably ...
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2answers
44 views

Characterisation of limit points of subsets of Hausdorff spaces

The theorem which i want to show is the following: For a Hausdorff space $X$ and a subset $A$ of $X$, $x$ is a limit point of $A$ if and only if every neighborhood of $x$ contains infinitely many ...
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1answer
51 views

Show that the set $\{(x, y) \in \Bbb R^2 : x > 0$ and $x^2 - y^2 = 1\} $ is path connected. [closed]

Show that the set $\{(x, y) \in \Bbb R^2 : x > 0$ and $x^2 - y^2 = 1\} $ is path connected. Please help me define a function.
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0answers
16 views

Hahn Banach theorem and supporting hyperplane theorem

The question is out of Rudin Functional analysis Chapter 3 problem 1. Call a set $H \subset \mathbb{R}$ a hyperplane if there exists real numbers $a_1,\ldots, a_n, c$ (with $a_i \neq 0$ for at least ...
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2answers
107 views

Where surjectivity goes in?

Let $X$ be an infinite set with the cofinite topology, and $f: X \to X$ a surjective function. Prove that $f$ is continuous if and only if $f^{-1}(\{x\})$ is finite for all $x\in X$. I know that $f$ ...
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1answer
70 views

Finding a denumerable set $X_0$ satisfying a condition.

Let $(X,\tau)$, with $X$ an uncountable set, $x_0 \in X$ fixed, be the space with topology generated by the collection: $$\mathscr{B} = \{ \{x\} \mid x \in X \setminus \{x_0\}\} \cup \{ A \subset X ...
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0answers
21 views

Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
1
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1answer
44 views

If a space $X$ has no isolated points, then nor does any dense subset of $X$.

$X$ is a topological space with no special properties. If a space $X$ has no isolated points, then show the same for any dense subset of $X$. Thanks a lot.
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0answers
28 views

Show that if $\forall$ $b\in B$, $f(b) \in T_y$ implies $f$ is open.

Let $f:(X,T_x) \rightarrow (Y,T_y)$ be a map between two topological spaces. Let $B$ be a basis for $T_x.$ Show that if $\forall$ $b\in B$, $f(b) \in T_y$ then $f$ is open. I am just looking for a ...
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3answers
46 views

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ is path-connected.

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ defined by $U_\epsilon(A) := \{x \in \Bbb R^n : d_A(x) < e\}$ is path-connected. If ...
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2answers
67 views

Every local property for $\mathbb{R}$ (any Connected Separable Space) holds globally?

I'm given this problem : Prove that "being polynomial" is a local property, meaning if $f: ℝ → ℝ$ is a polynomial in a neighborhood of each real point, then $f$ is a polynomial. I think I ...
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1answer
20 views

properties of connect sets in plane

If $A\subseteq\Bbb{R^2}$ is connect and is not single point,then proof $A\subseteq A^\prime$.and give counterexample that $$\\(cl(A))^°=A^°$$ is not true.note that $cl(A)$ is closure of $A$
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2answers
50 views

Accumulation point(s) of $\mathbb{R} \setminus \mathbb{N}$ in $\mathbb{R}$

Find the accumulation point(s) of $\mathbb{R} \setminus \mathbb{N}$ in $\mathbb{R}$ Let $A = \mathbb{R} \setminus \mathbb{N}$. We will denote our accumulation point(s) by $A^\prime$. I have a ...
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1answer
53 views

Is $\{(x,y) \in \mathbb R^2 : xy=0 \}$ homeomorphic to $\mathbb R$?

Is $\{(x,0) : x \in \mathbb R \} \cup \{(0,y) : y \in \mathbb R \}$ homeomorphic to $\mathbb R$ ? I am totally stuck and I don't even have any intuition whether they should be homeomorphic or not . ...
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0answers
13 views

Weak-* closure and convexity

I'm trying to write a proof of Goldstine's theorem : the weak-* closure of the unit ball of a normed vector space $X$ is the unit ball of the second dual $X^{**}$. At some point I would like to use ...
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1answer
20 views

Equivalence relation, product and quotient spaces

I have a problem with the following: "Define a relation $\sim$ on $R^2$ by $(u,v) \sim (x,y)$ if and only if both $u-x$ and $v-y$ are integers. Show that for each point $(x,y) \in R^2$ there exists ...
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1answer
69 views

If $f:(0,1)^n \rightarrow (0,1)$ is continuous, is $( x_1 ,\ldots,x_n) \mapsto (x_1 f(x_1,x_2,…,x_n),\ldots,x_n f(x_1,x_2,…,x_n))$ an open map?

Suppose $f:(0,1)^n \rightarrow (0,1)$, where $n>1$, is a continuous function. Define a function $g : (0,1)^n \to (0,1)^n$ by $$g(x_1,x_2,...,x_n):=(x_1 f(x_1,x_2,...,x_n),\ldots,x_n ...
0
votes
1answer
33 views

Explicit construction of retraction for Brouwer's fixed point theorem (disk)

So I'm trying to prove the Brouwer fixed-point theorem for the disk, arguing by contradiction and using the theorem that states that there is no retraction from the unit disk $D^2$ to the unit circle ...
0
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1answer
38 views

Show that if $X$ and $Y$ are regular, then so is the product space $X\times Y$.

Show that if $X$ and $Y$ are regular, then so is the product space $X\times Y$. I have no idea how to prove that. Even for a simplest example (the Standard topology) I can't make a point in ...
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2answers
28 views

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected.

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected. Here I use the following criterion for $X$ to be connected: A metric space $(X,d)$ is ...
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1answer
17 views

Exist a homeomorphic copy of $\mathcal{C}$ contained in $f(X)$. Where $\mathcal{C}=2^\mathbb{N}$.

If $X$ be a nonempty perfect Polish space, $Y$ a second countable space, and $f.X\to Y$ be inyective and Baire measurable. Question: Exist a homeomorphic copy of $\mathcal{C}$ contained in $f(X)$. ...
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1answer
44 views

Help creating a more insightful proof looking at closures of a metric space

My lecture notes from my metric space course contained the following practice questions. I am getting very confused by this question because I found the following statement on wikipedia "A metric ...
5
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3answers
319 views

mapping homotopic to the identity map

Please give me a hand with this problem, It was on my exam, and I just couldn't solve it. Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is ...
2
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1answer
401 views

Proving a topological space is/is not Hausdorff?

I know this is a basic question, but I am having trouble proving that particular topological spaces are/are not Hausdorff and was wondering if I could get some guidance. For example, I have to decide ...
2
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1answer
38 views

Proving an equivalent statement for the Stone-Weierstrass theorem

In my metric space course, we were taught the Stone-Weierstrass theorem as follows We were told however that the second condition ("A contains the constant functions") may be replaced by the ...
2
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1answer
37 views

Conifolds and Exotic Spheres

First of all, a disclaimer: I'm a physicist trying to understand mathematical aspects of some solutions I've encountered while studying string theory, and am certainly not a mathematician of any sort. ...
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1answer
24 views

Cardinality of an open dense set in a compact Hausdorff space

Let $\kappa$ be an infinite cardinal and let $X$ be a compact Hausdorff space of size $2^\kappa$. Let $U$ be a dense open subset of $X$. Can you give a lower bound for the cardinality of $U$. ...
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2answers
127 views

Reverse of Jordan curve theorem

Let $K$ be a compact subset of $\mathbb R^2$ such that $\mathbb R^2\setminus K$ is not connected. Is it true that $K$ contains a simple closed curve?
5
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1answer
88 views

Finding a good cover such that its lifting is still a good cover

Let $Y$ be a compact manifold, $X$ a topological space and $f: X \to Y$ a surjective map. Suppose further that every point in $Y$ has arbitrarily small open neighbourhoods such that their preimages in ...
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1answer
35 views

Is a closed convex set $E$ in $\mathbb{R}^n$ equal to the closure of its interior?

I have seen similar questions relating the interior of a convex set equal to the interior of its closure. However, I can't find anything that says if $E$ is a closed convex set, then $E = ...
0
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1answer
15 views

Isolated points of the split interval

Let us consider the split interval $S(I)$: that's the product space $I\times 2$ endowed with the lexicographic order. If we take the Cantor space $2^\omega$, why $S(2^\omega)$ has countably many ...
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0answers
22 views

Verifying a subbasis of the weak topology

So, I am working on a problem and I have shown that for $0 < p < 1$, $(\ell^p)^*= \ell^\infty$, I am now asked to show that the set of all $x$ with $\sum |x(n)| < 1$ is weakly bounded, but ...
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1answer
29 views

Question about convergence in a metric space

For part a) my strategy was showing that since E is sequentially compact, by the Borel-Lebesgue theorem it is compact. For part b) I am not sure how to solve the problem. Can I simply use the ...
4
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1answer
37 views

Question from a topology textbook regarding the uniform topology

I am able to prove that T maps continuous functions to continuous functions but I am lost when it comes to proving T is linear. I am also unsure how to solve b) and c). Thanks for any help you can ...
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1answer
47 views

Definition of $X \times_Y X$

Let $X,Y$ be topological spaces and let $f: X\to Y$. Questions: What is $X\times_Y X$? What is the map $\Delta_f: X \to X\times_Y X$?
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1answer
92 views

compact convergence topology

If $Y$ is a topological space, $Z$ a metric space. We can define the topology of compact convergenc and we have the compact open topology on $Z^Y$. Why are these two the same topology on $Z^Y$ ? ...
3
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3answers
131 views

Why are convex polyhedral cones closed?

Let $V = \mathbb{R}^n$, $v_1, \dots, v_s \in V$ and let $\sigma = \text{Cone}(v_1, \dots, v_s) = \{r_1v_1 + \dots + r_sv_s \mid r_i \geq 0\}$ be the associated convex polyhedral cone in $V$. Why is ...
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1answer
21 views

$f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , is the graph of $f$ connected in $\mathbb R^2$?

Consider the function $f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , then $f$ is not continuous on $\mathbb R$ . Is the graph of $f$ i.e. $G(f) :=\{ ...
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2answers
17 views

Stabilisers of group action open imply the action is continuous

Let $\mu \colon X \times G \longrightarrow X$ be the action of a topological group on a set $X$. We consider $X$ to be a topological space with the discrete topology. Suppose that for all $x \in X$, ...
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0answers
57 views

$A \subset \Bbb R$ such that $A$, $clA$, $int(A)$, $cl(int(A))$, $int(clA)$ are pairwise distinct

Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^0$ , $(\bar A)^0$ , $\overline{A^0}$ are pairwise distinct ? I found an example from a book that such a ...
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0answers
19 views

To an orbifold $\tilde{s}/\Gamma$, if $\Gamma\prime$ is a subgroup of $\Gamma$, then if $\tilde{s}/\Gamma\prime$, is also an orbifold? [closed]

Here $\Gamma$ is the group generated by the side pairing transformations. I guess it's not true, but I can't find a counterexample, could you give me some ideas? Thank you!
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0answers
34 views

Showing the Sum of $n-1$ Tori is a Double Cover of the Sum of $n$ Copies of $\mathbb{RP}^2$

I want to show that the non-orientable surface of genus $n$ has a 2-sheeted cover by an orientable surface of genus $n-1$. The base cases are easy: $S^2$ covers $\mathbb{RP}^2$ and I worked on a ...
3
votes
3answers
640 views

Is the cartesian product of homeomorphisms again a homeomorphism?

If we have two homeomorphisms $f:A\to X$ and $g:B\to Y$, then is it true that $f\times g:A\times B\to X\times Y$ defined by $(f\times g)(a,b)=(f(a),g(b))$ is again a homeomorphism? I think the answer ...
0
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2answers
44 views

Show that there is a $N \in \mathbb{N}$ such that $x_n \in A$ for all $n \geq N$.

Question: Let $A$ be an open subset of $\mathbb{R}^p$. Suppose than $(x_n)$ is a sequence in $\mathbb{R}^p$ such that $x_n \to x$, where $x \in A$. Show that there is a $N \in ...
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3answers
65 views

Clarification of proof in Munkres' Topology book

Theorem 20.3 on Page 123 of $\S$ 20 in J. Munkres' topology book is followed immediately by a proof. Within the proof, there is a line that I am having trouble understanding. First the Theorem: ...
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3answers
36 views

Why does closedness and boundedness for $S = \{ v \in V : || v||_{\infty} = 1\}$ imply that $S$ is compact in a finite dimensional vector space $V$?

Suppose $V$ is a finite dimensional vector space, with basis $\{e_1, \ldots, e_n\}$, over the reals $\mathbb R$ and $|| v ||_{\infty} = || a_1 e_1 + \ldots + a_n e_n||_{\infty}=\max_{1 \le k \le n} ...