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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Questions about a topological category

Given topological spaces $(X_i,\tau_i)$ with sets $\mathbf S_i=\{\mathcal A\in \tau_i^2|\mathcal A\supseteq\Delta X_i^2\}$, where $\tau^2$ is the product topology and $\Delta$ is the diagonal. The ...
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Subbases and half-planes

If $(X,d)$ is a metric space, it's easy to show that $H(x,y)=\{w\in X\mid d(x,w)>d(y,w)\}$ is open in the topology $\tau$ induced by $d$. Is, in general, $\{H(x,y)\mid (x,y)\in X\times X, x\neq ...
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Two disjoint connected and bounded open sets in the plane that shares the same boundary

In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary. My attempt: Since both my sets need to be bounded, my ...
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54 views

Dimensions definition?

Apparently dimension is "informally" defined as "the minimum number of coordinates needed to specify any point within it". For example we need at least 3 numbers to describe any point in the 3D space ...
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Sphere in finit dimensional space

please How to prove that the injection $i: S^{m-1}\rightarrow \mathbb{R}^m$ is homotopic to a constant ? Where $S^{m-1}=\{x\in \mathbb{R}^m, |x|=1\}$ Thank you.
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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 ...
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1answer
18 views

Regular spaces, weight and dense subsets

It is known that in case a space $X$ is regular ($T_1$ + $T_3$), its weight is less than or equal to $2^{d(X)}$, where $d(X)$ is the density of $X$. What is an example of a Hausdorff space for which ...
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50 views

Why does the antipodal on $S^2$ have degree $-1$?

I'm reading the post here by Arthur to explain that there is no smooth vector field on $S^2$. I don't understand it very well: The simplest I can remember off the top of my head is this: ...
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96 views

Is the inverse limit of simplicial maps between finite directed graphs also a graph?

I think I have an intuitive understanding of why the following statement might be true, but I am not sure how to go about proving it. It's also possible my intuitive understanding is wrong and the ...
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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 ...
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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 ...
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38 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, ...
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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 ...
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2answers
48 views

An example of a Lindelöf topological space which is not $\sigma$-compact

I am looking for an example of a Lindelöf topological space which is not $\sigma$-compact. I have looked in Counterexamples in Topology, but, if I am not wrong, all the examples there which meet my ...
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Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip?

Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity ...
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1answer
243 views

Properties of Topological Groups

I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve. The book defines a topological group as a group equipped with a topology where for ...
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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?
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2answers
291 views

closed set in another closed set

I'm new here and I'm hoping that maybe I could get some help with something my teacher told me. He said that it is possible to have a closed set nested within another closed set where the intersection ...
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1answer
72 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 ...
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1answer
21 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?
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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 ...
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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 ...
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1answer
53 views

TVS: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ...
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2answers
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TVS: Topology vs. Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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0answers
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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 ...
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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 ...
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0answers
64 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 $X$, ...
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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 ...
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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
36 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 ...
2
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1answer
63 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
2
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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 ...
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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\}, ...
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2answers
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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: ...
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1answer
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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 ...
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0answers
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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, ...
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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 ...
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2answers
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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 ...
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1answer
355 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in ...
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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 ...
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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)?
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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 ...
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2answers
58 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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Baire sets in locally compact Hausdorff spaces

(This is a follow-up to Compact $G_\delta$ subsets of locally compact Hausdorff spaces.) Suppose $X$ is a locally compact Hausdorff space. The Baire sets in $X$, denoted by $\mathcal Ba(X)$, comprise ...
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1answer
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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
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1answer
68 views

How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
3
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2answers
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$[0,1]\times[0,1]$ 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 ...
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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 ...
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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 ...
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$E$ 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 ...