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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7 views

If $A$ is a subset of a topological space, then $A' \subseteq A$ versus For any closed subset $A$ of a topological space, $A' \subseteq A$.

I need to determine which of the following are true and prove it... if it is false then I have to give a counterexample. If $A$ is a subset of a topological space, then $A' \subseteq A$ versus ...
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0answers
8 views

Vector bundles and isomorphism

I have $E \mapsto X$ which is an n-dim vector bundle, and sections $ s_1,..,s:n : X \mapsto E$ such that for any $x \in X$, the elements $s_1(x),..,s_n(x)$ are linearly independent in the vector ...
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1answer
28 views

True or false: sets, subsets, and topologies in $\mathbb R$

I am pondering the following statements about sets, subsets and topologies in $\mathbb R$. The empty set is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$. Any open ...
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1answer
23 views

show that$ X$ is uncountable if $(X,d)$ is connected [duplicate]

if $(X,d)$ is a connected metric space and there exist non-constant real-valued continuous function $f$ in $X$, show that $X$ is uncountable
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1answer
28 views

Visualizing weird topology

Let $(\mathbb R, T)$ be a toplogical space. A set $U$ is open iff for each $p \in U$ there is an open interval $I_p$ such that $p \in I_p$ and every rational number in $I_p$ is in $U$. The problem is ...
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0answers
21 views

Path-components of the general linear group using only elementary algebra

Let $E(c)$ be an elementary matrix of the type to add $c$ times a row to another row when applied to another matrix on the left (with $c$ in some off-diagonal position $(i, j)$), and, with the usual ...
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2answers
44 views

show that $\mathbb{Z}$ is totally disconnected

Show that $(\mathbb{Z},d)$ is totally disconnected (where $d$ is the metric induced by the Euclidean metric on $\mathbb{R}$). I think that to prove this I should use contradiction but not really sure ...
3
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1answer
38 views

Is $X = \{0,1,…,9\}^{\mathbb N}$ totally disconnected?

Consider X with the product topology and $\{0,1,...9\}$ with the discrete topopogy. I already attempted to prove and disprove it, both without success, so I'd be happy to know whether it is true or ...
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2answers
14 views

finding $C^1$ path on an open and path connected set.

Given an open and path connected set $U\subseteq \mathbb R^n$, is there a way to find a $C^1$ path between every $a,b\in U$? If so, is there a general proof of existence of such path?
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0answers
29 views

semicontinious functions in topology space [on hold]

Problem is: Let $X$ be a topologicyl space.. Prove that: A function $f\colon X\to\mathbb R$ is continuous function if and only if $f$ is lower semi-continuous and upper semi-continuous. The ...
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1answer
12 views

Nets and sequences in a 1st couuntable space.

Let $(X,\mathcal{T}$) be a 1st countable topological space. Let $(x_\delta)_{\delta\in\Delta}$ be a net converging to $x$. Does there exist a sequence $(x_n)$ that converges to $x$ and which is a ...
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0answers
33 views

Proving a function is continous only if a set is open

FACT: A function is continuous if and only if the inverse image of every open set is open. Now, suppose we have $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous everywhere and lets say we have an ...
6
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3answers
279 views

If a set is compact then it is closed

Show that if a set is compact then it is closed. definitions: Let $A\subset \mathbb{R}$. A point $p\in\mathbb{R}$ is an accumulation point or limit point of $A$ if and only if every open set $G$ ...
3
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1answer
20 views

Difference between continuous and uniformly continuous functions on a dense metric subspace.

Let $X$ be a dense subset of metric space $(\tilde X,d)$. Let be $(Y,d')$ be a complete metric space and $ f: X \rightarrow Y$ a continuous mapping. It follows from density that for all points in ...
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0answers
29 views

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$? [duplicate]

Is $\{\frac{1}{n}:n\in\mathbb{N}\}$ nowhere dense in $[0,1]$ for the metric induced from the Euclidean metric on $\mathbb{R}$? I think that yes, it is nowehere dense because ...
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0answers
16 views

Interpretation of a weakly compact set of functions

I'm having trouble really grasping the idea of a weakly compact set. The set I have under consideration is a set of functions $M_c$, where $$M_c=\{f:W(f)\leq c\},$$ where $c\geq0$ and $W(f)$ is a ...
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1answer
50 views

Do finite products commute with colimits in the category of spaces?

Let $X$ be a topological space. The endofunctor $\_\times X$ of the category of all topological spaces does in general not possess a right adjoint, since the category is not cartesian closed. Is it ...
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0answers
16 views

Which of the following are proper patches. (Showing that an inverse of a mapping is continuous)

In which of the following cases is the mapping $\mathbf{x}:\mathbb{R^2} \to \mathbb{R^3}$ a proper patch? (a)$\mathbf{x}(u,v)=(u,uv,v)$ (b)$\mathbf{x}(u,v)=(u^2,u^3,v)$ ...
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1answer
15 views

open set in subspace

Could someone check the following? Consider $M=\{0\} \cup \{ 2^{-k}: k \in \mathbb{N}\}$ a subspace of $\mathbb{R}$. Which of the subsets $\{0\}$ and $\{2^{-k}\} (k\in \mathbb{N})$ are open in ...
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1answer
15 views

Relation between open divisible subgroup and the quotient of the group with subgroup

I wanted to prove the following proposition: Let H be an open divisible subgroup of an abelian topological group G. Then G is topologically isomorphic to H x G/H. As for the proof, using extension of ...
2
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2answers
32 views

a simple question about topological space

X is a topological space, $ A\subseteq B\subseteq X $, if $A$ is a nowhere dense subset of $B$ , then $A$ is a nowhere dense subset of the whole space $X$? Is this right? I thind it's right, but I ...
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0answers
27 views

Every point in a Tychonoff Space is contained in a compact set

This should be very elementary, but I just can't see it: Is every point in a Tychonoff Space contained in a compact set? I tried to look for a counterexample and figured that it cannot be locally ...
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1answer
23 views

Closedness of $\{ x \in 2^A : x(\neg p) = \neg x(p) \}$ for a Boolean algebra $A$ and $p \in A$

I'm reading Matthew Dirk's The Stone Representation Theorem for Boolean Algebras, and am trying to follow the proof of Proposition 3.4 on p.6: Proposition 3.4. Let $A$ be a Boolean algebra, and ...
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1answer
35 views

Number of Hausdorff topologies on a set with $100$ elements.

Find the number of Hausdorff topologies on a set with $100$ elements. I know that number of topologies on a set with $2$ elements is $4$, with $3$ elements is $29$ , with $4$elements is $355$ , ...
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1answer
32 views

Connected Sets Examples

(a) Give an example of a connected set $A \subset \Bbb R^n$ such that $\Bbb R^n\setminus A$ is not connected. (b) Give an example of a compact set $K \subset \Bbb R^n$ which is not connected. So far ...
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1answer
28 views

Projection Mappings are Quotient Mappings?

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X=X_1\times X_2$ is a product space, then the first coordinate projection is a ...
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0answers
25 views

When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
5
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4answers
73 views

Prove that if $A$ is both open and closed, $A=\mathbb R$. [duplicate]

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
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0answers
17 views

Necessary and sufficient condition for an orthonormal system to be total

Let $H$ be a Hilbert space over a field $\mathbb K$. Prove that an orthonormal system $\{a_n\}_{n=1}^{\infty}$ in $H$ is total if and only if: $\forall$ $x \in H$, the following holds: ...
2
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1answer
20 views

Image of continuous injective map has empty interior.

Let $\varphi :\left [ 0,1 \right ]\rightarrow \mathbb{R^2}$ be a continuous injective map. Let $I = \varphi \left ( \left [ 0,1 \right ] \right )$ be the image of this map. Prove that $I$ has empty ...
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1answer
23 views

Dense sets and Empty Interior

if $A$ is dense in $X$, is there a relation which shows in which cases $A$ has empty interior ? $\mathbb{Q}$ has an empty interior as a dense set in $\mathbb{R}$, so does its complementary in ...
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0answers
17 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
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1answer
28 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
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1answer
21 views

Example 4, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to figure out saturated open sets?

Let $X$ be the closed unit ball $$ \{ \ x \times y \ \colon \ x^2 + y^2 \leq 1 \ \} $$ in $\mathbb{R}^2$, and let $X^*$ be the partition of $X$ consisting of all the one-point sets $\{ \ x \times y ...
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1answer
41 views

Find two disjoint open sets $U, V$ such that $A\subseteq U, B\subseteq V$ where $A,B$ are closed.

Let $A, B$ be two disjoint closed subsets of a certain metric space $(M,d)$. Show that there exist disjoint open subsets $U, V \subseteq M$ such that $A\subseteq U, B\subseteq V$. Give ...
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1answer
26 views

Prob. 3, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How is this map a quotient map that is neither open nor closed?

Let $\pi_1 \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be projection on the first coordinate. Let $A$ be the subspace of $\mathbb{R} \times \mathbb{R}$ consisting of all points $x \times y$ ...
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1answer
28 views

I think $A,B$ must be closed and disjoint

Prove that in every metric space, $(X; d)$, is possible find a continous function$f\colon X\to \mathbb{R}$ such, if $ A $ and $ B $ are two subsets of $ X $ then $ f(x) = 1 $, for every $ x\in A $ ...
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0answers
15 views

Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
2
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1answer
11 views

Hausdorff spaces for continuous bijections

I have the following question being posed: Suppose $f:X\rightarrow Y$ is a continuous bijection. Prove that if $Y$ is Hausdorff, then $X$ is also Hausdorff. Here's my attempt: Consider any $a,b\in ...
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1answer
24 views

Find a suitable counterexample?

Is the following statement true or false? If a sequence $(x_n)$ with an infinite range $\{ x_n : n \in \mathbb{N} \}$ has precisely one accumulation point, then $(x_n)$ converges. I know the ...
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3answers
28 views

Show that the closed ball is closed in $\mathbb{R}^p$

Let $r>0, p \in \mathbb{N}$ be given. Show in detail that the closed ball $\{ x \in \mathbb{R}^p : ||x|| \leq r \}$ is closed in $\mathbb{R}^p$. Let $A = \{ x \in \mathbb{R}^p : ||x|| \leq r ...
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0answers
48 views

Id proof this quastion [on hold]

Id proof this quastion pleace tomorw to me
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2answers
22 views

Examples of path component maps

I understand what needs to be done for the first part, i have to somehow map $1$ point onto $1$ point, in a map where there exists $2$ points... so the inverse map is injective, but how is this ...
4
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3answers
78 views

Mistake in (Baby) Do Carmo? Elementary topology of surfaces.

If you have the book, it's proposition 2 of section 5.3. If not, the proposition reads: Given any two points p and q $\in$ a regular, connected surface S, there exists a parameterized piecewise ...
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0answers
16 views

$\operatorname{Fr}( p(\overline U) )$ where $p : X \to Y$ is a closed, not necessarily continuous, surjection, and $U \subset X$ is open

Question: Let $p : X \rightarrow Y$ be a closed (not necessarily continuous) surjection. If $U$ is open, then $$\operatorname{Fr} ( p(\overline U) ) \subset p(\overline U ) \cap p(X - U).$$ I ...
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2answers
54 views

Example 1, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to verify that this map is closed?

Let $X$ be the subspace $[0,1] \cup [2,3]$ of $\mathbb{R}$, and let $Y$ be the subspace $[0,2]$ of $\mathbb{R}$. The map $p \colon X \to Y$ defined by $$ p(x) \colon= \begin{cases} x \ &\mbox{ ...
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0answers
18 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
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0answers
12 views

Finite intersection of arbitrary union not stable for arbitrary unions

It is a set-theoretic exercise to prove that the set of arbitrary unions of finite intersections of sets is still stable under finite intersections. However it is not true that finite intersection of ...
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1answer
20 views

A proof related to diameter of a simplex S

Question: Prove that the diameter $\mathcal p(S)$ of a simplex $\mathcal S$ equals the greatest Eucledian distance between two vectors in the simplex. My opinion: We all know what every vector in the ...
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0answers
11 views

Existence of real valued function continuous at $\mathbb Q$ discontinuous at $\mathbb R\backslash \mathbb Q$ [duplicate]

Does there exist a real-valued function of a real variable which is continuous at every rational point and discontinuous at every irrational point?