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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1answer
19 views

Double cover of the real projective plan $RP^2$

We know that the double cover of the real projective plane is a sphere $S^2$. How can we visualize this? I am thinking of consider the cross cap model of the projective plane. So start by drawing the ...
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24 views

Question about a cycle when doing topological sorting

First off, this is a homework question, but I'm just a bit confused on some of the smaller details of doing a topological sort. The homework question can be seen here (it shows the graph). It says: ...
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2answers
64 views

What are “finiteness” and “discreteness” when it comes to compact sets?

I recently found this answer by Qiaochu Yuan but I'm not sure what "finiteness" and "discreteness" function are in the context of compactness. I've read What does it mean when a function is finite? ...
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1answer
30 views

Showing that $A$ irreducible $\Leftrightarrow \bar{A}$ irreducible

Let $A$ be a subspace of a topological space $X$. Show that $A$ is irreducible $\Leftrightarrow$ $\bar{A}$ is irreducible. My attempt: $\Leftarrow$) Suppose that $A$ is reducible. Then, ...
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3answers
199 views

How do you prove the set $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed? [duplicate]

Let $f : \mathbb R \to \mathbb R$ continuous. Prove that graph $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed. I'm a little confused on how to prove $G$ is closed. I get the general strategy is ...
3
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1answer
37 views

Measure of $E$ is the limit of the measure of the open set $\mathcal{O}_n$

Suppose $E$ is a given set, and $\mathcal{O}_n$ is the open set: $$\mathcal{O}_n=\{x : d(x,E) < \frac 1n \}.$$ Show: (a) If $E$ is compact, then $m(E)=\lim_{n \to \infty} ...
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1answer
60 views

Can $\mathbb{R}$ be homeomorphic to some $X \times X$? [duplicate]

I got this question in my topology exam and I had no idea how to make it, topology can be so weird sometimes it's hard to imagine some spaces. My heart told me that it is not possible, but I do not ...
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0answers
41 views

Equivalent topology definitions

Given any two absolute values on an arbitrary field k, one defines the absolute values to be equivalent if they define the same topologies. I am having trouble understanding how the following ...
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1answer
67 views

Neighborhoods and Metric Spaces in Real Analysis

In my analysis class, we are looking at metric spaces and topologies. We were asked to prove the following two theorems. I believe I have the first one completed, but am somewhat confused by the ...
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1answer
38 views

product topology in subspaces

Let $A_i$ be an arbitrary subspace of $X_i$. Show that the product topology on $\prod_i A_i$ is equal to the relative topology on $\prod_i A_i$ as a subset of the product space $\prod_i X_i$. The ...
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1answer
33 views

Prove that for a dense set $A$ and an open set $U$, $U\subset \overline{A\cap U}$

In a topological space $X$ a subset $A$ is called dense if $\overline{A}=X$. Show that for a dense set $A$ and an open set $U$ $U\subset \overline{A\cap U}$ My solution $U$ is open $\Rightarrow$ ...
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0answers
21 views

Continuity of Product Topology [duplicate]

Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a ...
0
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1answer
29 views

Sequence and convergence with subsequence

Prove:$x_{n}\rightarrow x \implies x_{n_{i}} \rightarrow x$ for any subsequence. Normally I would have a proof of this at least attempted proof, but I am really confused what my professor is trying ...
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1answer
26 views

$A$ is trivially closed if and only if $X\backslash A$ is trivially open

A set $U$ of topological space $X$ is called Trivially open if $U=$Int$(\overline{U})$ and set $C$ of topological space $X$ is called Trivially closed if $C=\overline{IntC}$ Prove that $A$ is ...
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1answer
23 views

Additional properties of closure and accumulation point

prove or disprove: $(\overline{\overline{A}}) = \overline{A}$ proof: This is true, since $\overline{A}$ is closed and $A$ is closed I am not sure if this is right, any suggestions would be greatly ...
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2answers
21 views

More properties of closure and accumulation point

Prove: $(\overline{A\bigcap B}) = \overline{A}\bigcap \overline{B}$ proof: Let $p\in(\overline{A\bigcap B})$ be an accumulation point of the intersection. First, let's assume that $p\in A\bigcap B$, ...
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2answers
74 views

Why do we need surjectivity in this theorem?

In class we proved the following theorem: Given $X_1,X_2$ ordered sets. Then any surjective increasing $\phi: X_1 \to X_2$ is continuous wrt the interval topology on $X_1$ and $X_2$. I was asked to ...
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0answers
26 views

Translation of GENUS to Portuguese

Does somebody know a translation (to portuguese) for "genus" in topology? Theorem: A nonsigular projective curve in $\mathbb{P}_2$ is topologically a sphere with $g$ handles. Definition: This number ...
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0answers
38 views

A set $U$ is open iff it is union of open balls

Let $(X,d)$ be a metric space. Consider the collection $\mathcal{T} = \{ U \subset X: \forall u \in U, \exists r>0 \; \; , B_r(u) \subset U \}$. We showed that $(X, \mathcal{T} )$ is a topological ...
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2answers
84 views

Examples of Separable Spaces that are not Second-Countable

In this post I give an example of a separable space that is not second-countable. What are other good examples?
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3answers
65 views

Topologizing Borel space so that certain functions become continuous

Let $X$ and $Y$ be compact metric spaces. Let $f:X \to Y$ be a Borel measurable map and suppose that $T:X \to X$ is a homeomorphism. Can one change the topology on $X$ such that $X$ is still a ...
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0answers
23 views

Intersection of Decreasing closed sets

If $\{E_n\}_{n \in N}$ is a sequence of closed nonempty and bounded sets in a complete metric space $(X,d)$, if $E_n \supset E_{n+1}$ and if $$\lim_{n \to \infty} \operatorname{diam} E_n = 0,$$ then ...
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1answer
34 views

When can you drop an inequality term when you have more than two?

I am working on a problem: $d$ and $d'$ are metric equivalents on a set $X$, meaning there exist $n > 0, n' > 0$ such that for all $x, y \in X$, $d(x,y) \leq n \cdot d'(x,y)$, $d'(x,y) \leq ...
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1answer
92 views

If $A_1,A_2,…$ is a sequence of subsets of a topological space. Prove following

If $A_1,A_2,...$ is a sequence of subsets of a topological space. Prove $\overline{\bigcup_{k=1}^{\infty}A_k} = \bigcup_{k=1}^{\infty}A_k \cup ...
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2answers
35 views

Irreducible topological spaces and irreducible Hausdorff space

A topological space $X$ is said to be irreducible if it cannot be written as a union $X = A∪B$, where $A$ and $B$ are proper closed subspaces of $X$ I want to try the following: Show: $(i)$ That ...
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0answers
32 views

What kind of topology is this (if it involves non-set operations)

Suppose we have $\cup, \cap$ defined as binary operations on "subsets" of some set $S$, where "subsets" can be something exotic like substrings of a string. Suppose also that they satisfy analogous ...
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2answers
48 views

Maximal space and Zorn's lemma

Let be $(X, \tau)$ a Hausdorff space without isolated points. I want to prove that for the order relation $\subseteq$, every order subset of $A =\left\{\tau_x: \tau_x \text{ is a topology on } X, \ ...
2
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1answer
33 views

Understanding proof that if $c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$ then $d_1$ and $d_2$ are topologically equivalent metrics

Theorem. If there are strictly positive constants $c_1$ and $c_2$ such that $$c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$$ for all $x,y \in X$, then $d_1$ and $d_2$ are topologically ...
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2answers
44 views

Equivalent properties for a locally compact space

I have this exercise: Let $(E,\tau)$ be a Hausdorff space; prove that the following are equivalent: 1) $(E,\tau)$ is locally compact. 2) For all $x\in E$ and $U\in \mathcal{V}_x$ an ...
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2answers
197 views

Four Color Theorems: Graphs vs. Maps [closed]

This question has changed dramatically from its original form. Please See the improved question. ORIGINAL QUESTION: There are two variants of the four color theorem that are commonly cited: (4CTG): ...
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0answers
19 views

Clarification needed for an example about Hausdorff space

This is from Kolmogorov & Fomin Introductory Real Analysis[p.86]. Example 4. Consider the closed unit interval $[0,1]$, where neighborhoods of any point $x\neq 0$ are defined in the usual way ...
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1answer
103 views

Utility of the 2-Categorical Structure of $\mathsf{Top}$?

It's well known that $\mathsf{Top}$ is a 2-category with homotopy classes of homotopies as 2-arrows. I'm a bit afraid to ask this question, but what is the utility of this 2-categorical structure? ...
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0answers
28 views

The “Wiggle Room” intuition for cofibrations

Often enough - for instance in the answer to this question - I have encountered the idea that an inclusion $i:A\subset X$ is a cofibration if $A$ has enough "wiggle room" in $X$. Although I have a ...
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3answers
31 views

Property of Closure and the limit point

Definitions: $A'$ is the set of all accumulation or limit points. $\bar{A} = A \cup A'$ - this is known as the closure of $A$. Let $A$ be a subset of $\mathbb{R}$. A point $p\in\mathbb{R}$ is an ...
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4answers
49 views

Embedding vs continuous injection (in topological vector spaces)

When working with topological vector spaces (say $X,Y$), the term “embedding” is often used for a continuous injection $f:X\rightarrow Y$. Now, $f$ is of course a bijection onto its image, but it's ...
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1answer
82 views

Hatcher's example of a pair without the HEP

At the bottom of page 15, Hatcher gives the pair $(I,A)$ where $A= \left\{1,0,\frac 12,\frac 13,\dots \right\} $ as an example of a pair without the HEP. He then attributes the problem to the failure ...
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2answers
31 views

Topology induced by norm

What is the meaning of topology induced by norm. To me topology is a collection of subsets satisfying certain rules. How can a norm induce a topology...? For example how can $\|\cdot\|_{2}$ induce a ...
3
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5answers
122 views

Problem book on general topology [duplicate]

Can one please suggest some problem book on general topology? I have Munkres but exercises are easy, that's why I want a book which consists of slightly non trivial or interesting problems. Also you ...
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0answers
31 views

Monodromy Theorem and homeomorpihsm

If (A,f) is a smooth unlimited covering surface of X, f maps A onto X, and X is simply connected, then the Monodromy theorem implies f is a homeomorphism? I can't see this totally. Do you have any ...
0
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1answer
16 views

Need to prove this lemma in order to classify isometres in $\mathbb{R}$

I am trying to use this lemma to prove that if an $f \in Isom_o(\mathbb{R})$, then $f(x)=x$ or $f(x)=-x$.. . The Lemma I want to use says: Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an isometry ...
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2answers
39 views

Help with understanding the concept of paracompactness.

The definition for paracompactness that I'm given is as follows: "A set $U\subseteq\epsilon$ is paracompact with respect to the topology ($\epsilon$,$\tau$) if and only if every open cover {$O_i$} of ...
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4answers
882 views

Why is an infinite dimensional space so different than a finite dimensional one?

In functional analysis there is a big difference between finite- and infinite-dimensional vector spaces. I have found other questions with nice answers here and here. However, I don't grasp the ...
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2answers
43 views

Product of two topologies?

Is the Cartesian product of two topologies again topology? According to my knowledge , NO ? So how the product of two or many topologies defined ? Could someone make it clear for me. I am new to ...
2
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1answer
65 views

Is every homeomorphism a quotient map?

Let $X$ and $Y$ be topological spaces. Let $f: X \rightarrow Y$ be a bijections such that $f(U)$ is open in $Y$ if and only if $U$ is open in $X$. Then $f$ is a homeomorphism and an open map. In ...
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1answer
27 views

Conneted normal space having more that one point is uncountable [duplicate]

A connected normal space having more that one point is uncountable. As I apply Uryshon Lemma to resolve this fact. Thanks.
1
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1answer
32 views

Additional properties of closure [duplicate]

Definitions: $A'$ is the set of all accumulation or limit points. $\bar{A} = A \cup A'$ - this is known as the closure of $A$. Let $A$ be a subset of $\mathbb{R}$. A point $p\in\mathbb{R}$ is an ...
4
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1answer
38 views

Applications of the fact deformation retracts are closed under pushout

Fact: Suppose we have a pushout diagram $$\require{AMScd} \begin{CD} A @>{f}>> C\\ @V{i}VV @VV{j}V\\ B @>>{g}> D\end{CD}$$ where $i$ is the inclusion of a deformation retract. ...
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1answer
45 views

Question of regular open

A set $U$ in a topological space $X$ is called regular open if $U=\text{Int}\left(\overline{U}\right)$. Similarly, a set $F$ is regular closed if $X\setminus F$ is regular open or equivalently ...
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4answers
58 views

Properties of Closure

Definitions: $A'$ is the set of all accumulation or limit points. $\bar{A} = A \cup A'$ - this is known as the closure of $A$. Let $A$ be a subset of $\mathbb{R}$. A point $p\in\mathbb{R}$ is an ...
0
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0answers
15 views

Hausdorff space and continuos functions [duplicate]

Lets $f,g:(X,\mathcal{G})\to (Y,\mathcal{H})$ continuous where $Y$ is Hausdorff and $D$ dense in $X$. If $f(x)=g(x)$ for all $x \in D$ then $f=g$. a suggestion please.