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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0answers
15 views

software for drawing sequence in metric space

Use what kind of software to draw a grapf describing open cover and a sequence in a metric space? For example, I need to show some subsets of an open cover and a sequence consists of many points in a ...
0
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1answer
14 views

Question concerning one rule of the topological calculus in terms of the interior operator

Define a topological space as a set $X$ and a function $\text{int}()$ assigning to every set $A\subseteq X$ the set $\text{int}(A)\subseteq X$ such that: (i) $\text{int}(A\cap ...
0
votes
1answer
20 views

Zero sets in completely regular spaces

I'm wondering if I am missing something from this portion of a problem (14.C.1 - Willard) A zero set in a topological space $X$ is a set of the form $f^{-1}(0)$ for some continuous ...
2
votes
2answers
64 views

$[0,1]\times[0,1]$ stays connected after removal of an interior point

I am self-studying Topology's connectedness and came across this simple problem: Show that $[0, 1] \times [0, 1]$ stays connected if you remove an interior point. Visually it looks simple ...
0
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2answers
43 views

How to find a continuous function that demonstrates that the set $\{(x,y):y>x\}$ is open?

Consider the set of points $U$ in $\Bbb{R}^2$ that lie above the line $y = x$, i.e. points $(a,b)$ such that $b>a$. Prove that $U$ is open and connected. The method that is recommended is showing ...
-1
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0answers
18 views

Convex basis and conical basis (how to draw?)

There is a question, I'm struggling with: Find vertices of the following described polyhedron, $P:=P(A,b)=conv(V)+cone(E)$ where $V$ is the set of all vertices of $P$ and $E$ is the set of all ...
0
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2answers
32 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 ...
1
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0answers
22 views

$E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering

Let $q:E\to X$ be a covering map. Then $E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering. My question is regarding the $"\implies"$ direction: If $E$ is compact, then ...
0
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1answer
21 views

Tychonoff spaces with small weight

Let $\kappa$ be an infinite cardinal. Is there a Tychonoff space $(X,\tau)$ such that $|X| = 2^\kappa$ and $(X,\tau)$ has weight $\kappa$ (= a basis consisting of $\kappa$ elements)?
4
votes
2answers
34 views

Two-sheeted covering of the Klein bottle by the torus

Prove that there is a two-sheeted covering of the Klein bottle by the torus. OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows: Then there are ...
2
votes
1answer
37 views

If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
6
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1answer
95 views

Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
3
votes
1answer
32 views

Discrete subspace of $\mathbb{N}^\mathbb{N}$

Endow $\mathbb{N}$ with the discrete topology. Does $\mathbb{N}^\mathbb{N}$ contain a discrete subspace of size $2^{\aleph_0}$?
1
vote
3answers
48 views

Is it possible to get a neighborhood with only finitely many points in it, in an infinite set?

If we have an infinite set, is it possible to find a neighborhood around a certain point in the set that has only finitely many points in it?
0
votes
0answers
28 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
3
votes
3answers
140 views

What's the definition of a “local property”?

Is a property called local if and only if for every point there exists a neighbourhood for which the property is true? For example: Let $X,Y$ be topological spaces. Then $f: X \to Y$ is continuous if ...
1
vote
1answer
25 views

Does this definition of “limit point” really work

I am reading Tapp's introduction to matrix groups for undergraduates. He gives the following definition of limit point of a set: A point $p \in \mathbb R^m$ is called limit point of a subset $S ...
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0answers
19 views

What is connexity (in simple language)? [on hold]

Please explain the meaning of connexity, connex elements in simple language
0
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1answer
19 views

Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
2
votes
0answers
49 views

A more detailed, rigorous proof that a suspension space is not necessarily contractible

Is my answer/proof correct? Please help me make my proof more rigorous and detailed. I need everything to be absolutely clear. Question: Let $X$ be a topological space. The suspension of $X$, ...
2
votes
0answers
14 views

Probability Density Function for Randomly Oriented Ellipse

I have an ellipse with a long aspect of a and a short aspect of b. The equation for this ellipse is found on this post: What is the general equation of the ellipse that is not in the origin and ...
6
votes
1answer
61 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
3
votes
1answer
73 views

Is my Proof Correct and Rigorous: Proving that Quotient Space is Hausdorff

Question: Let $X$ be a topological space and let $A ⊂ X$. Define an equivalence relation $∼$ on $X$ such that the equivalence classes are: • $A$ itself, and, • Singletons {$x$} such that $x /∈ A$. ...
2
votes
2answers
45 views

All neighborhoods of a compact subset of an open space are subsets of that open space

Let $K$ be a subset of $U$, with $K$ compact and $U$ open. Prove that there is an $\epsilon > 0$ such that for all $p$ in $K$, a neighborhood of radius $\epsilon$ of $p$ is a subset of $U$. Note: ...
0
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1answer
35 views

In Hausdorff spaces, compact sets are closed [duplicate]

I've been stuck on this exercise for awhile. It was suggested that I fix a point in the compact set, but I didn't know what to do with that. Can anyone provide an answer?
1
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1answer
28 views

Complement Topology on $S^3$

Given $S^3$ the three dimensional sphere and it's usual Euclidean topology, call it $\tau$, consider $U:=\{(X \setminus A) \mid A \in \tau \}$. Does $U$ form a topology on $S^3$? My guess is no. ...
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0answers
37 views

Is a topological space locally compact in this case? [on hold]

If $X$ is a topological space for which every point is contained in a locally compact subspace then is $X$ also locally compact? Many thanks in advance.
5
votes
2answers
69 views

Is this a criterion for continuity?

Given a topological space $(X,\tau)$ and the product space $(X^2,\tau_2)$. Define the diagonal $\Delta X^2=\{(x,x)\,|\,x\in X\}$ and a set $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta ...
4
votes
3answers
98 views

Show $ \{ (\xi,\eta,\zeta) \in \mathbb{R^3} : \xi = \eta = \zeta \}$ is closed

Show $ F =\{ (x_1,x_2,x_3) \in \mathbb{R^3} : x_1 = x_2 = x_3 \}$ is closed. I'd like help finishing off my solution below. Other answers are appreciated as well. It suffices to show that the ...
8
votes
6answers
528 views

What does “removing a point” have to do with homeomorphisms?

I am self-studying topology from Munkres. One exercise asks, in part, to show that the spaces $(0,1)$ and $(0,1]$ are not homeomorphic. An apparent solution is as follows: If you remove a point, ...
1
vote
1answer
44 views

If $Y$ is a locally compact topological subspace of $X$ then is $X$ also locally compact?

If I have a topological space $X$ and a subspace $Y$ which is locally compact then does this mean that $X$ is also locally compact? For local compactness, I want to show that every point has a ...
1
vote
0answers
28 views

Is ${\cal B}=\{ (p_1,q_1)\times (p_2,q_2)\times …| \, p_i-q_i=p_j-q_j \}$ a basis for product topology?

There is a problem from my topology course. For a collection ${\cal B}=\{ (p_1,q_1)\times (p_2,q_2)\times ...| \, p_i-q_i=p_j-q_j \, \forall i,j \in \mathbb{N}\}$. Prove or Disprove: ${\cal B}$ is a ...
1
vote
1answer
15 views

Continuity of evaluation maps in the topology of compact convergence on $C([0,\infty),\mathbb{R}^{n})$

I'm trying to prove that the evaluation maps $e_{x}:C([0,\infty),\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ given by $e_{x}(f):=f(x)$ are Lipschitz-continuous with respect to the metric ...
1
vote
1answer
29 views

Non-connectedness in the plane

Let us have open $V \subset \mathbb{R}^2$ and $x\in V$. Now, how can I prove that the quotient space $V\backslash\{x\}$ is not simply connected? Pictorially I understand it as the failure of a loop to ...
2
votes
1answer
22 views

Quick question: Map being smooth vs Graph being submanifold of the product space [on hold]

Is $f:X\rightarrow Y$ smooth if and only if the graph $\Gamma_f$ is a closed submanifold of $X\times Y$? Thank you very much.
6
votes
5answers
337 views

concepts which is present in metric space but not in topological space

I want to know some concepts which is present in metric space but not in topological space. The one that first comes to mind is uniform continuity, equicontinuity i.e. concepts defined with some kind ...
1
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1answer
39 views

$\mathbb{N}$ is a Compact Space with the Co-finite Topology?

Let $X$ be the topological space on the set $\mathbb{N}$ with the cofinite topology. I am having a hard time seeing why this is compact in the topological sense. If each open $n$-hood on $X$ ...
4
votes
1answer
41 views

Quasi Cauchy sequences in general topology?

Suppose $(X,\tau)$ is a topological space and that $(X^2,\tau_2)$ is the product space. Now define $\mathscr S\!_\tau=\{W\in\tau_2|\Delta X^2\subseteq W\}$, where $\Delta X^2=\{(x,x)|x\in X\}$, ...
1
vote
1answer
30 views

Theorem on continuity and closed sets

In the text Mathematical Analysis, Second Edition by Tom Apostol theorem 4.24 states: Let $f:S\rightarrow T $ be function from the metric space $(S,d_{S})$ to another $(T,d_{T})$. Then $f$ is ...
1
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1answer
26 views

Determining a finite subcover for a compact topological space

Suppose (X,τ) is a compact topological space and C ⊆ X is closed. Show that C is compact in (X,τ) So far I have come up with: Let P be an open cover of C. Thus C is closed, then its complement, P', ...
1
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1answer
31 views

If two non-disjoint subsets are connected, why does their union have to be connected?

So X and Y are two sets such that their intersection is nonempty. I want to show that if X and Y are each connected, together their union is connected. I tried proving this by contraposition and I've ...
2
votes
1answer
67 views

Prove that these loops are homotopic [duplicate]

Let $G$ be a topological group with identity element $e$. Let $f,g: (S^1, (1,0)) \to (G,e)$ be loops in $G$ with base point $e$. We define $f * g: (S^1, (1,0)) \to (G,e)$ by $$f * g(s) = f(t) \cdot ...
2
votes
0answers
34 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
8
votes
1answer
80 views

Topological Hangman

Suppose a mysterious adversary has captured me and challenged me to the following topological game. We fix some finite set, say, $X = \{1, 2, 3, \ldots, n\}$, and my adversary secretly constructs a ...
2
votes
1answer
28 views

Unit close disc to prove a matrix algebra identity?

I need to prove that every $3 \times 3$ matrix with real positive entries has one eigenvector with a positive eigenvalue. Now, how do I prove this using the fact that the set $B=\{x=(x_1,x_2,x_3)\in ...
1
vote
1answer
45 views

The intersection of infinitely many dense open sets may not be dense

I know that if $A$ and $B$ are dense open subsets of topological space $X$, then $A\cap B$ is also dense open. Furthermore, if $X$ is a Baire space, then the countable intersection of dense open ...
0
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0answers
15 views

A basic doubt on upper semi-continuity of set-valued maps

Upper Semi-Continuity for set valued maps have two definitions $h:\Bbb R^d \to 2^{\Bbb R^d}$ is upper semi-continuous if 1) Sequential definition : $x_n \to x$, $y_n \to y$ and $y_n \in h(x_n)$ ...
0
votes
2answers
28 views

Continuity definition

The definition of continuity I have always seen is that if $X,Y$ are topological spaces and $f : X \rightarrow Y$, then $f$ is continuous if for any open $B \subseteq Y$, $f^{-1}(B)$ is open in X. ...
3
votes
2answers
50 views

A weaker Hausdorff topology on $\mathbb R$ with different system of compact subsets?

Consider the real line $\mathbb R$ with the usual topology, generated by intervals $(a,b)\subseteq{\mathbb R}$. Do there exist a weaker Hausdorff topology on $\mathbb R$ with different (a wider) ...
7
votes
5answers
152 views

Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...