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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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?
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1answer
45 views

Section 22 in Munkres' TOPOLOGY, 2nd edition: How to establish this equivalence?

Let $X$ and $Y$ be topological spaces; let $p \colon X \to Y$ be a surjective map. Then $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is ...
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20 views

Prob. 4, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to decide which cases to consider?

We need to show that the ordered square satisfies the first countability axiom. I'm not able to decide as to which separate cases to consider. By definition the ordered square is the product $I ...
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1answer
32 views

Is a limit point compact subspace of a Hausdorff space necessarily closed? [on hold]

I think the answer should be "no", but I can't give a counter-example.
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24 views

If $d(x_0,y_j)\to d(x_0,y_0)$, then $y_j \to y_0$.

Consider a metric space $X$, and a compact subset $C\subset X$.Let $x_0\in X-C$. We can show that there is a point $y_0\in C$ such that $d(x_0,y)=\inf_{y\in C} d(x_0,y)$. Now suppose there is ...
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14 views

generalize the question every every intersection of nested sequence of compact non-empty sets is compact and non-empty

I'm aware how to prove that the intersection of nested sequence of compact non-empty sets is compact and non-empty. but I want to generalize this question to transfer the hypothesis of having nested ...
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2answers
25 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
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14 views

weak closure of unitary group in $B(H)$

Let $H$ be a Hilbert space with dim $H=\infty$ , and $\cal{U}$ be the group of all unitaries on $H$. Show that the weak closure of $\cal{U}$ is a semigroup with respect to the multiplication. I know ...
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2answers
56 views

Quotient Maps and Compact Hausdroff Spaces

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X$ and $Y$ are compact Hausdroff space and $f:X\rightarrow Y$ is a continuous ...
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1answer
23 views

What does continuity of a function mapping a topological space to a real line interval mean?

It makes sense for continuity to be defined on a function mapping a real line to a real line. Or how continuity is defined on a function between two topological spaces (every preimage of an open set ...
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3answers
210 views

Does every cover have an irredundant subcover?

While composing an answer for this question, I got troubled by a technical point. I wanted to assert the existence of an irredundant subcover of a given open cover, but realized I'm not sure how to ...
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2answers
89 views

Why is indiscrete topology unmetrizable?

For instance, the indiscrete topology for $X$ cannot arise from a metric when $X$ has more than one point. One way to see this is to note that the complement of a one-point set in a metric space is ...
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1answer
21 views

Condition on closures implies discreteness of topology.

I'm supposed to prove that if $(X, \tau)$ is a topological space such that $\overline{A \cap B} = \overline{A} \cap \overline{B}$ for all $A,B \subset X$, then $\tau$ is the discrete topology. My ...
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38 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
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2answers
54 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
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2answers
24 views

The image of a path-connected set under a continuous map is path-connected

Show that if $X$ is path-connected and $f:X\to Y$ is a continuous map, then the image $f(X)$ is path-connected. In order to show this is path connected I know the definition is :
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1answer
27 views

Show that the finite complement topology is connected

I am looking at $\mathbb{R}^n$ with the finite complement topology and need to show it's connected. I know that a connected doesn't have any non-trivial clopen sets. For $U \in T$ where T is the ...
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1answer
25 views

Topology show X is compact

I no the following where we can use the definition of compact to be:
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1answer
26 views

A partition of the unit square such that the quotient space is the Klein bottle

Write down a partition $X^*$ of the unit square $X=[0,1]\times[0,1]$ such that the quotient space is the Klein bottle. I understand the definition of Quotient topology and Partitions, however, ...
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2answers
37 views

Topology without tears exercises 1.2 #6 i)

Let T be a topology on a set X such that T consists of precisely four sets; that is , $T = \{X, \emptyset, A, B\}$, where $A$ and $B$ are non empty distinct proper subsets of $X$. Prove that $A$ and ...
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0answers
28 views

Solution Verification: Prove in detail that the open rectangles in the Euclidean plane form an open base

I want some verification and/or some polishing on my proof. However if it is good, please let me know (I think this is highly unlikely to happen). Problem. Prove in detail that the open rectangles ...
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1answer
37 views

Extending a topology to linear combinations?

Suppose I have a topological space $X$, and some arbitrary field $K$. I am trying to nicely describe a set of functions on ${}_K X$, the set of $K$-linear combinations of values in $X$. I feel like ...
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1answer
14 views

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set.

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set. I am in an introduction to proofs class. I have to decided if this is a true ...
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2answers
44 views

How to show that there exists a sequence in $[0,1]$ such that the set of accumulation points of the sequence is $[0,1]$

This is related to homework but I am trying to find a special case first and see if I can generalize it. The problem is to construct some sequence $(x_n)$ in $[0,1]$ such that the accumulation points ...
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1answer
12 views

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$.

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$. I think this is a true statement and I therefore need to prove ...
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1answer
30 views

Why is this a convex polygon?

Let $\text{E}(2)$ be the group of isometries of the plane $\mathbb R^2$. Then $\text{E}(2)=\text{O}(2)\times\mathbb R^2$ as a topological space and is the semi-direct product as groups. Let $G$ be a ...
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0answers
33 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...
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1answer
28 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
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2answers
21 views

On convergent sequences

Suppose that i have and open and surjective map between two metric spaces $\pi\colon X\to Y,$ and a sequence $(x_n)_{n\in \mathbb{N}}$ such that its image by $\pi$ converges. Is it true that ...
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25 views

Properties of a Connected set in $\mathbb R ^n$

Prove that: If $E$ is an open connected set in $\mathbb R ^n$, let $A\subseteq\mathbb R^n$ be both open and closed with respected to $E$, then $A=E$. Intuitively it seems to be true because if $A$ ...
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1answer
27 views

Compactness and convergence

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this ...
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0answers
36 views

How to show a map is a homeomorphism?

I have calculated two of the properties of homeomorphism. Where I have found the bijective mapping and showed that $f$ is continuous. However i am not sure how to show that $f^{-1}$ is continuous?
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Metric Spaces open function [on hold]

Let X and Y are two metric spaces and f is bijection function from X to Y . Prove that f is open iff f is closed
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3answers
32 views

Topology space closed set [on hold]

Why $A =\{ (x,1/x), 0\neq x\in R \}$ is closed in $R^2$?
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1answer
28 views

The special unitary group SU(2) is homeomorphic to the 3-sphere

The special unitary group is defined as $$SU(2) = \{A\in M_{2,2}((C) \mid A\bar A^T=I\}$$ Show that this is homeomorphic to the $3$-sphere $$\mathbb{S}^3 = \{(a,b\in \mathbb{C}^2\mid ...
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1answer
29 views

Topological structure/graph from a paper

This question is based off a paper titled "On designing heteroclinic networks from graphs." I'm having a difficult time visualizing something "drawn in 4-dimensions" projected down to a 2-dimensional ...
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1answer
32 views

Join of closed embeddings is a closed embedding

An exercise from James's book General Topology and Homotopy Theory asks the reader to prove that if $\phi_1:X_1 \to Y_1$ and $\phi_2:X_2 \to Y_2$ are closed topological embeddings, then $\phi_1 * ...
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1answer
25 views

Difficulty with a differentiation of measures proof

This shows up in a proof about differentiating measures. I'm having trouble figuring it out: For any $x \in \mathbb{R}^n$, let $\mathcal{C}_r(x)$ denote the set of open cubes with diameter less than ...
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4answers
99 views

How should I think of an open vs. closed set?

I've been studying introductory topology for a little bit now. I came across this video which explains open sets in a way I have never thought of. Even though the video is pretty elementary, I didn't ...
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2answers
44 views

show the function is a homeomorphism?

Show that $f:(-1,1)\to\mathbb R, f(x)=\frac{x}{1-x^2}$ is a homeomorphism. In order for this function to be a homeomorphism it needs to be bijective and its inverse needs to exist and be ...
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0answers
33 views

Topology, maps, continuity

I know how to write all the maps, however, how would i find out which of those are continuous.
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1answer
46 views

Let $g: S^2 \to S^2$ be continuous and $g(x) \neq g(-x)\ \forall x$. Prove that $g$ is surjective.

Let $g: S^2 \to S^2$ be continuous and $g(x) \neq g(-x)\ \forall x$. Prove that $g$ is surjective. The hint that if $p \in S^2$, then $S^2 - \{p\}$ is homeomorphic to $\mathbb{R^2}$. It is ...
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13 views

projective plane and topology [duplicate]

Gluing the Mobius strip with a disk altogether along their boundaries gives $\mathbb RP^2$. Please give me some hint or explanation how to solve it, why it is true. And give a topology on this ...
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0answers
20 views

What is the Topology of Cantor set as the subset of real line with standard euclidean topology? [duplicate]

It may be an odd question. But, I particularly want to know the open's of cantor set with inherited topology of real line.
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1answer
19 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
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1answer
34 views

Question on Furstenberg topology on Z and P subspace of primes

Hi all I was given this question: I have Z (the integers) with the Furstenberg topology on it, i.e. the topology induced by non constant arithmetical progressions presented here, and I am asked to ...
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0answers
50 views

transformation of a folded piece of paper!!!

This is a question in the book Real Mathematical Analysis by Charles Chapman Pugh and I don't know how to face it! : Fold a piece of paper in half. (a) Is this a continuous transformation of one ...
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1answer
77 views

How many topologies exist on a finite set?

In my topology class we are asked to list all topologies on a $3$ element set. I have found $29$ and this should be the correct result. Now I wonder whether there is some formula that determines this ...
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2answers
57 views

Proving that if a set is both open and closed then it is equal to the real numbers

Prove that if $A$ is both open and closed then $A = \mathbb{R}$ also as one suggested let $A \neq \emptyset$ You may use what ever definition of open and closed you would like, just avoid going into ...
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15 views

Homomorphisms between countable spaces and Euclidean spaces?

Is there some place to start reading about homomorphisms between countable (discrete) spaces and Euclidean spaces or $l_2$? I know it is a rather general question, but I am not sure what I am looking ...