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Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

4
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107 views

Existence of a Zero of a Continuous Map

Suppose that $f: \mathbb{R}^n \to \mathbb{R}^n$ is a continuous map, satisfying $$ \langle x,f(x) \rangle \geq 0, \forall x\in S^{n-1}=\{x\in \mathbb{R}^n: \Vert x\Vert =1 \} .$$ Prove that there ...
3
votes
3answers
67 views

How to prove or disprove the following set is compact?

$P$ is set of all real polynomial in one variable. Define $$d(p,q)=\max\{|p(x)-q(x)|:x \in[0,1]\}$$ and $$K=\{p \in P,d(p,0)\leq 1 \}$$Prove or disprove $K$ is compact. I think the only way to prove ...
2
votes
3answers
71 views

Proving that $\{ x \in \mathbb{R}^n : |x| = 1, x\geq 0\}$ is homeomorphic to $\overline{B_1(0)} \subset \mathbb{R}^{n-1}$

I want to show that $A= \{ x \in \mathbb{R}^n : |x| = 1, x\geq 0\}$ is homeomorphic to $\overline{B_1(0)} \subset \mathbb{R}^{n-1}$, where $\overline{B_1(0)}$ is the closed unit ball. I was thinking ...
2
votes
3answers
333 views

Cluster Point vs Limit Point

Are there any differences between a cluster point and a limit point? If yes, is every cluster point is also a limit point?
2
votes
3answers
93 views

Is the following subset of $\mathbb{R}^2$ open, closed or neither?

I am given the subset $A=\{(x, \sin(1/x)) | x>0\} \bigcup \{(0,y) | y \in [-1,1] \}\subset \mathbb{R}^2$ equipped with the standard Euclidean metric I came to the conclusion that it is not open ...
2
votes
3answers
39 views

$X$ is c-closed iff every countably compact subset of $X$ is closed

I need help to understand definition of c-closed. In here https://www.sciencedirect.com/science/article/pii/0166864180900279, author said that $X$ is c-closed iff every countably compact subset of $X$ ...
2
votes
3answers
160 views

Topology of the torus

Theorem: There is no open covering $T^k=U_1\cup...\cup U_k$ of the $k$-torus such that the map $$H_1(V,\mathbb{Z}) \rightarrow H_1(T^k,\mathbb{Z}) $$ has rank at most $(i-1)$ for every component $V$ ...
2
votes
3answers
309 views

Constructing a compact set with countably infinite many limit points

I have an exercise to construct a compact set with countably infinite many limit points. I am trying to use the set: $$A = \{0\} \cup \{\frac{1}{n}: n=1,2,3,\ldots \}\cup \{\frac{1}{n}+ \frac{1}{m}:...
2
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3answers
72 views

Infinite Intersections in Topology

I'm having trouble finding an example that a professor wants. The problem is as follows. Find an example of a topological space $(X,\mathcal{T})$ with open sets $U_{i}\in\mathcal{T}$, $i\in\mathbb{...
2
votes
3answers
82 views

Existence of a topology satisfying a certain condition

This might be a really silly question but i'm curious to know if the following is true: Let $X$ be an infinite set with topologies $\tau_1$ and $\tau_2$ such that $\tau_1\subset\tau_2$. Is it always ...
2
votes
3answers
49 views

Subset of discrete and closed subset is also discrete and closed?

I came upon the following question (probably trivial...): Let $X$ be a topological space Suppose $N \subseteq M \subseteq X$ and $M$ is closed and discrete in $X$. Then is it true that $N$ is ...
2
votes
3answers
157 views

Is division by two continuous in topological groups?

Assume that $(G, +)$ is a Hausdorff topological abelian group which is uniquely divisible by two, i.e. the function $x \to 2x = x+ x$ is a bijection. Clearly, it is also continuous. My question is if ...
1
vote
3answers
31 views

How to prove this property on interior and closure of sets?

$\newcommand{\inr}{\operatorname{int}}$ Assume that $(X, \tau)$ is a topological space, and $ A \in P (X) $. Show that $ \inr(A) \neq \emptyset $ if only if for all $B \in P(X)$ with $\overline{B}=...
1
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3answers
75 views

Best resources for learning/practicing beginning topology?

I'm an undergrad with minimal experience in proof-based classes, and I'm in a pickle with a current course. Normally, I have trouble understanding and/or retaining information during lectures, so I ...
1
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3answers
86 views

Prove that if for $E$ and $F$ connected sets, if $E \cap F \neq \emptyset \rightarrow E \cup F$ is connected

Prove that if $E \cap F \neq \emptyset \rightarrow E \cup F$ is connected. I am trying to do it by contradiction. Assume that $E \cup F$ is disconnected. Therefore, $\exists U, V$ open, such that $$...
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3answers
70 views

Prove continuity from preservation of sequential limits but do not use $f(\bar{A}) \subseteq \overline{f(A)}$

Munkres Topology: Theorem 21.3. Let $f : X \rightarrow Y$. If the function $f$ is continuous, then for every convergent sequence $x_n \rightarrow x$ in $X$, the sequence $f(x_n)$ converges to $f(x)$...
1
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3answers
55 views

Given a closed disc staying in an open set in $\mathbb{C}$, is there a larger open disc within the open set which contains this closed disc?

Let $U$ be open and $D \subset U$ be a closed disc. In the book Complex Analysis by Stein (P.39 Corollary 2.3), he claimed in the proof that there exists a slighly larger open disc $D' \subset U$ such ...
1
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3answers
73 views

The set $A=\{0\} \cup \{\frac 1n \mid n \in \mathbb N\}$

For the set $A=\{0\} \cup \{\frac 1n \mid n \in \mathbb N\}$, I understand that $\{\frac 1n \mid n \in \mathbb N\}$ is open and closed in $A$ because it is a union of all the connected components $\{\...
1
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3answers
129 views

When is a topological property useful for deciding if two spaces are homeomorphic

I know that being connected is a topological property. But, my question is what are examples of two spaces where this isn't helpful? Is this simply two spaces that are disconnected or having some ...
1
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3answers
78 views

Showing a space is not Hausdorff

A space is Hausdorff if given two different points $x$ and $y$, there are two open disjoint sets $G_x$ and $G_y$ such that $x\in G_x$ and $y\in G_y$. Prove that if $K$ is compact and $K$ is not closed,...
1
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3answers
66 views

Prove $x^2-y^2<z^2$ is open

I'm trying to prove $x^2-y^2<z^2 $ is open. I believe this is a cone shape, can I simply assert that $∀ x ∈ P$ there exists $r > 0$ such that $B_r(x) ⊆ P$, where $P$ is my subset? Or is there an ...
1
vote
3answers
151 views

order topology and continuous functions

Let $X = [0,1], Y = X\times X$. $X$ with the standard topology of $\mathbb{R}$ and $Y$ with the order toplogy, where $(x,y) < (a,b)$ if $x < a$ or $x = a$ and $y < b$. Let $f:X\...
1
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3answers
126 views

Definitions of the 'limit points'

I knew that the definition of limit points $x$ of a subset $E$ of $\mathbf{R}^d$ is : For any open ball $B$ containing $x$, $(B\setminus \{x\})\cap E\ne\emptyset$ Or something like these arguments ...
0
votes
2answers
227 views

Convex Homotopy

Suppose $f , g : X \to U \subset \mathbb R^2$ are two mappings from a topological space $X$ to a convex set $U$. Prove that $f$ and $g$ are homotopic, using only the definition of the product mapping....
0
votes
2answers
73 views

Conditions for continuity of composition

Let $f:X\rightarrow Y$, $g:Z \rightarrow Y$, and $h:Z \rightarrow X$ be functions such that $f \circ h = g$, where $X, Y$ and $Z$ are topological spaces. What are some weak conditions that we must ...
0
votes
2answers
121 views

If there exists an open set $U$ in $X$ such that $A = Y \bigcap U$ then $A $ is open in $Y$

Let $Y$ be subspace of a metric space $X$. Show that $A \subset Y$ is open in $Y$ if and only if there exists an open set $U$ in $X$ such that $A = Y \bigcap U$. My Try: Let $A$ be open in $Y$. Then ...
0
votes
2answers
860 views

Showing set $A = [0,1)$ in a discrete metric is open and closed

Let $X = \mathbb R$ with metric $ \rho_0 : X \times X \to \mathbb R$ defined by $$ \rho_0 (x,y) := \begin{cases} 1, & x \ne y;\\ 0, & x = y; \end{cases} $$ - $\bullet$ If $A = [0,1)$, then ...
0
votes
2answers
112 views

Let $F$ be a closed subset of $(X,d)$ and suppose that $\inf\{d(x,z): z \in F\} = 0$ for some $x \in X$. Prove that $x \in F$.

All the limit points of $F$ are contained in $F$ since $F$ is closed. $d(x,z)$ is a set of real numbers. It is obvious that $\inf\{d(x,z): z \in F\} = 0$ for some $x \in X$. I have no idea how to ...
0
votes
2answers
92 views

Boundary Sets are Closed

The following question is from Fred H. Croom's book "Principles of Topology" Prove that the boundary of a subset $A$ of a metric space $X$ is always a closed set. My attempt is as follows: ...
0
votes
2answers
58 views

Showing set is closed

Consider the set {$\frac{1}{2},\frac{1}{3},...,0$}. Show that this set is closed. How would one approach such a question without pursuing an epsilon-delta kind of proof?
0
votes
2answers
430 views

Proof check: proving a neighborhood is an open set?

I want to prove that a neighborhood is an open set by picking an arbitrary point in it and showing it's an interior point. On my final exam I couldn't think of a way to use the triangle inequality(...
0
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2answers
163 views

Wedge Sum Embedding with Inclusions

Let $X$ and $Y$ be two disjoint topological spaces, $x_0\in X$, $y_0\in Y$ and we consider the Wedge Sum (the quotient of the union by the relation $x_0\sim y_0$). I want to proof that $\pi \circ i_X$...
0
votes
2answers
42 views

Conditions on $A,B$ that are inherited by $A+B$.

Let $A,B$ be subsets of $\Bbb{R}$. Which of the following is false: If $A,B$ are bounded, then so is $A+B$. If $A,B$ are open, then so is $A+B$. If $A,B$ are closed, then so is $A+B$. If ...
0
votes
2answers
397 views

Definition of embedded in topology

One of the questions in my topology homework starts with: Suppose $G$ is a graph and $L \subset G$ is an embedded circle. I have looked around and found lots of definitions for an 'embedding' but ...
0
votes
2answers
100 views

The intersection of a descending sequence of open sets with a point in common has a nonempty interior.

If in a topological space $X$, we have a descending infinte sequence of open sets, $\{V_n\}_\mathbb N$ ($V_{n+1}\subseteq V_n$) such that for a fixed $z\in X$, $z\in V_n$ for each $n\in \mathbb N$. ...
0
votes
2answers
96 views

A question about “basis for a topology”

Let $(\mathbb{R}^2,\tau)$ a topological space where $\tau=\{\emptyset,\mathbb{R}^2\}\cup \{D_k\}_{k\in\mathbb{R}}$ such that $D_k=\{(x,y)\in\mathbb{R}^2,x+y<k\}$ How to prove that $B=\{D_k,k\in \...
0
votes
2answers
80 views

Show that the set $B := \left \{ (r,\infty) \mid r \in \mathbb{R}\right \} $ is a basis of some topology on $\mathbb{R}$, but not a topology itself.

Show $B := \left \{ (r,\infty) \mid r \in \mathbb{R}\right \} $ is a basis of some topology on $\mathbb{R}$, but not a topology itself. The definition of a topology basis is the following: A ...
0
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2answers
46 views

Why is this set connected?

I don't understand something about a proof of that if $X_i$ is a connected space for every $i\in I$, then $X=\Pi_{i\in I}X_i$ is connected. It is this: Let $x\in X$. Define $C$ the set of all $y\in X$...
0
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2answers
81 views

Is the boundary of this set compact?

Let $X$ be a topological space, $Y$ a subspace of $X$ and $A\subseteq Y $ such that $\partial(A)$ is compact in $X$. Is $\partial(A)$ compact in $Y$?
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votes
2answers
52 views

Can a realistic rubber mannequin be turned inside-out.

A realistic human-shaped mannequin is made of topological-grade rubber. All the body cavities that are accessible from outside without piercing any tissue, are faithfully reproduced, for example the ...
-1
votes
2answers
64 views

Brouwer's fixed point theorem for permutations

Brouwer's fixed point theorem: Let f:[a,b]->[a,b]. There exists some x in [a,b] such that f(x)=x. The above theorem deals with functions going from a set to the same set. These functions are, by ...
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votes
2answers
98 views

Topology: subbasis but not a basis

I'm trying to find a set of $\mathbb R^2$ that is a a subbasis, but not a basis for a topology on $\mathbb R^2$. I think I have a set, but am having trouble going about proving this. My set is $S=\...
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votes
2answers
127 views

How i can let it be a topology on $\mathbb{R}^2$

I have these sets $$\Omega_r=\{(x,y)\in \mathbb{R}^2, (x-1)^2+(y+1)^2\geq r^2\}, r\geq0$$ I know that the set $\tau$ defined by $\varnothing$ and all $\Omega_r$ is not a topology because$$ \bigcup_{...
-1
votes
2answers
42 views

Prove: if $A$ is a closed ball in $\mathbb{R}^n$ then $int(A)$ is an open ball

Let $A$ be a closed ball, $A=\{x:||x-a|\leq r\}$, Lets take $y\in A$ such that $\{y:||y-a||<r\}$ by definition all these elements of the group form an open ball $B$ How should I continue?
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votes
2answers
95 views

Finding the interior and exterior of this set

$Let$ $S$ be the set of $\frac{a}{b}$ such that $a,b$ are rational numbers. Q1) What is the interior of $S$? Rational number divided by a rational number will give me the rational number set. I ...
-1
votes
2answers
168 views

A theorem about one-dimensional convex sets

Suppose we have a non-empty convex set which does not consist of only one point such that it belongs to the same line, then this set is either a line segment(closed, half-open or open),a ray(closed or ...
-1
votes
2answers
47 views

It's given for $\left \{ A=\frac{1}{n},n\in \mathbb{N} \right \} $=${1,\frac{1}{2}…}$. Why is $0$ accumulation point?

It's given for $\left \{ A=\frac{1}{n},n\in \mathbb{N} \right \} $$={1,\frac{1}{2}....}$. Why is $0$ accumulation point? It's never going to reach 0 because it's not in $\mathbb{N}?$
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votes
2answers
28 views

Set is a bounded set

I have a question. I have to show that $$S1 = \{x \in \mathbb{R}^2 : x_1 \geq 0,x_2 \geq 0, x_1 + x_2 = 2\}$$ is a bounded set. So I have to show that $\sqrt{x1^2+x2^2}<M$. I have said that $M&...
-4
votes
2answers
83 views

Prove that $\Delta = {\{(x,x);x}$ in ${X\}}$ in $X\times X$ can be written as an intersection of open sets iff X is $T_1$.

Prove that $\Delta = {\{(x,x);x}$ in ${X\}}$ in $X\times X$ can be written as an intersection of open sets iff X is $T_1$. The topology in $X\times X$ is the product topology. $T_1$ is the first ...
-4
votes
2answers
149 views

Topology Spaces with onto open map.

0 down vote favorite Let $(𝑋,𝒯)$ and $(𝑋',𝒯\ ')$ be topological spaces. Suppose $𝑓: 𝑋 → 𝑋'$ is an onto open map that has the extra property that $𝑈′$ open in $𝑋'$ implies that inverse of $𝑓(...