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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6
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1answer
44 views

Topology and Measures

I apologize if this question is a bit vague; I'm just wondering if there is a concept like what I'm talking about, or if I'm just lost. I'll start with just some thoughts. I looked a bit, and I don't ...
3
votes
0answers
31 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
3
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0answers
40 views

Density of the rationals in the reals

While studying measure theory I have encountered the following set, $$U_\varepsilon=\bigcup_{n\in \mathbb{N}}(q_n-\varepsilon /2^n,q_n+\varepsilon/2^n),$$ where $(q_n)_{n\in \mathbb{N}}$ is an ...
4
votes
1answer
38 views

When does a continuous function defined on a non-compact closed and bounded convex set has a fixed point?

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
6
votes
1answer
65 views

inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
2
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1answer
55 views

elementary topology exercises reference

Can anyone recommend a good collection of elementary topology exercises? A pdf collection of undergraduate problem sets and homework, or midterm and final exams that I could practice on? Even a ...
3
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2answers
34 views

Directed sets to describe a topology with nets.

I'm studying some things related to ultrafilters on metric and topological spaces and trying to prove theorem in a general setting, so the following question came to my mind. Let $S$ be a ...
0
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1answer
19 views

Baire property for finite discrete spaces

Does it makes sense to assume that a nonempty open set of a finite discrete topological space has the Baire property?
0
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0answers
32 views

Bounded polyhedrons

Given a bounded polyhedron $P=P(A,b)$ and with $x$ s.t. $Ax<b$, show: $\exists \ \alpha>0 \ \ \ \ \ \text{ s.t.}\ \ \ \alpha^Tx\leq1, \ \ \ \ \forall x \in P $ How I should proceed to prove ...
0
votes
2answers
40 views

queston about T4- spaces

please can can anyone tell me that is it is true? i found this statement (question) in one old book of topology. i think it is just printing mistake. statement(question) is prove that every T4 space ...
3
votes
1answer
35 views

Simply connected and connected in complex analysis

Can some one please help me with this, why is third set in the picture not simply connected. The definition of simply connected (in space of complex numbers) is: A set is said to be simply ...
1
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3answers
35 views

Is ${\mathbb R}^n$ with the product topology the same as the metric topology

I have looked at several places into the definition for product spaces. Now all of the definitions I have seen, define the product space topology as generated from the product of sets $U_i$, for which ...
1
vote
1answer
29 views

A question about a perfect space and a linear order on it

Suppose I have a nonempty perfect Polish space $X$, and there's some linear order $<$ on it (it is not related to the topology on $X$ in any way). How can I prove that there is a point $y$ in $X$ ...
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3answers
41 views

Topological space examples for 1.compact but not Hausdorff and not connected. 2. not compact not Hausdorff not connected.

I made a table about topological spaces with or without connected, compact, and Hausdorff properties. However, I cannot find the example for compact+not Hausdorff+not connected and for not compact+not ...
2
votes
1answer
32 views

Covering map is proper $\iff$ it is finite-sheeted

Prove that a Covering map is proper if and only if it is finite-sheeted. First suppose the covering map $q:E\to X$ is proper, i.e. the preimage of any compact subset of $X$ is again compact. Let ...
0
votes
1answer
28 views

about finite discrete space

question : is all finite discrete spaces are $T_2$- space, $T_1$ -space and also $T_0$ -space. i have taken very simple example: X = {a, b, c} and topology $T = \{ ∅, X, \{a\}, \{b\}, \{c\}, ...
2
votes
0answers
43 views

Dual of path in a space.

Is there a notion dual to the notion of a path in a topological space? Given that a path in a space $X$ is a continuous function from the interval $[0, 1]$ to X, what would the dual of this notion be, ...
0
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0answers
27 views

software for drawing sequence in metric space [on hold]

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
votes
1answer
23 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
27 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 ...
3
votes
2answers
88 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
votes
2answers
56 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 ...
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votes
0answers
20 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
votes
2answers
33 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
32 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
votes
1answer
30 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
44 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
38 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
votes
1answer
103 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
34 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 $2^{\aleph_0}$?
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3answers
51 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
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0answers
38 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
146 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
28 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
20 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
23 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
62 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
68 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
78 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
49 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
votes
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
vote
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
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2answers
71 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
533 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
1answer
37 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
16 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 ...