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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107 views

Example for a set in $\Bbb R^p$ whose interior is $\emptyset$ and closure is $\Bbb R^p$

The following exercise was in the Elements of Real Analysis by Bartle. Give an example of a set $A$ in $\Bbb R^p$ such that $A^{\circ} = \emptyset$ and $A ^ - = \Bbb R^p$. Can such a set be ...
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2answers
32 views

Suppose $\forall p \in X, \, \exists f \in C(X,\mathbb{R})$ s.t. $f^{-1}(0) = \{p\}$. Show $X$ Hausdorff

An exercise from John Lee's Introduction to Topological Manifolds: Suppose $X$ is a topological space, and for every $p\in X$ there exists a continuous function $f:X \to \mathbb{R}$ such that ...
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2answers
49 views

Regular or normal topological space

How do we define an open neighborhood around a closed set? My question is with respect to a normal or regular topological space where we use the concept of an open neighborhood around a closed set.
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0answers
86 views

Difference between Metric Space and Topological Space

I am reading Chapter 11 of Real Analysis written by Royden and Patrick (4th). It says "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise ...
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2answers
36 views

$\Omega$ open and $K$ compact $\Rightarrow B_\varepsilon(0)+K\subset\Omega$ for $\varepsilon$ small enough.

Let $\Omega\subset\mathbb{R}^N$ be an open bounded set and $K\subset\Omega$ compact. I'm trying to prove that for $\varepsilon$ small enough we have $(B_\varepsilon(0)+K)\subset\Omega$. Consider the ...
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1answer
66 views

When is distance to the boundary always less than that to the exterior?

Let $X$ be a metric space with the distance function $d$. Given a subset $S \subseteq X$, what are the required conditions on $X$ and $S$ are so that $d(x, \partial S) \leq d(x, \operatorname{ext} S)$ ...
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2answers
46 views

Show that $f:X\to Y$ is an open mapping iff for all $A \subset Y, \, f^{-1}(A^\circ)\supset f^{-1}(A)^\circ$.

Struggling with a proof from basic topology: Show that $f:X\to Y$ is an open mapping iff for all $A \subset Y, \, f^{-1}(A^\circ)\supset f^{-1}(A)^\circ$. I can show the $\implies$ direction ...
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1answer
35 views

How to prove that if $x\in X$ and $y\notin\overline{ X}$ then $|[x,y]\cap\partial X|=1$?

Given $x,y\in\mathbb{R}^n$ let us define $[x,y]=\{(1-t)x+ty\in\mathbb{R}^n;\;t\in[0,1]\}$. Geometrically (in two or three dimensions), $[x,y]$ is the line connecting $x$ and $y$. Let ...
4
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2answers
81 views

Is there a continuous injective map from $\mathbb{R}$ that has compact image?

Suppose I have a function $f: \mathbb{R} \to X$, where $X$ is some non-compact metric space. Is it possible that $f$ is injective yet has compact image? If the answer is yes, what characterizes ...
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2answers
241 views

Manifold that is NOT smooth

Could someone provide an example of a manifold that is not smooth? All manifolds that come to mind are smooth! By a manifold, I mean a hausdorff, second countable locally euclidean space.
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1answer
54 views

Proof of equivalence of definitions T3 spaces

I'm trying to prove that the following two definitions of $T_{3}$ spaces are equivalent, and hitting an impasse. Definition 1: For any point and closed set not containing the point, there are ...
4
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2answers
74 views

If the function $\varphi \colon Z\rightarrow C(X,Y)$ is continuous then $F\colon Z\times X\rightarrow Y$, $F(z,x)=\varphi (z)(x)$ will be continuous.

If the function $\varphi :Z\rightarrow C(X,Y)$ ($C(X,Y)$ with compact-open topology) is continuous and $X$ is locally compact, then $$F\colon Z\times X\rightarrow Y$$ $$F(z,x)=\varphi (z)(x)$$ will be ...
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143 views

Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
2
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2answers
160 views

Let X be an infinite set with a topology T, such that every infinite subset of X is closed. Prove that T is the discrete topology.

Let X be an infinite set with a topology T, such that every infinite subset of X is closed. Prove that T is the discrete topology. I have somewhat of an answer but I don't think it's enough to prove ...
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2answers
137 views

Why is pointwise continuity not useful in a general topological space?

On page 27 of Lee's Introduction to Topological Manifolds, he writes In metric spaces, one usually first defines what it means to be continuous at a point...in topological spaces, continuity at a ...
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1answer
32 views

Theorem why are $X_n$ closed?

Our professor gave us the following theorem. Let $T$ be a compact topological space then any infinite subset of T has at least a limit point. Proof $T$-compact space. Suppose $B$ subset of $T$ with ...
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1answer
289 views

Let $(Z,T)$ be the set of integers with the cofinite topology. Find the limit points of the sets $A=\{1,2,3\}$ and $E$ (even integers)

Also, compute the closure and the interior of each set. I have tried proofs for finding the limit points, but I just want to check my reasoning is enough and having trouble with the closure and ...
2
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5answers
219 views

The distance between two sets inside euclidean space

Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be $$ d(A,B) = \inf \{ \|x-y\| : x \in A, \; \; y \in B \} $$ For any $A,B$, I want to prove that $d(A,B) = d( ...
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1answer
121 views

Question about Lipschitz function

Suppose $A = (a_{ij})$, $1 \leq i \leq m$, $1 \leq j \leq n$ is an $m \times n$ matrix. then $A: \mathbb{R}^n \to \mathbb{R}^m$ given by $$ A(x) = A(x_1,\dots,x_n) = \left( \sum_{j=1}^n ...
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2answers
33 views

question about norms of linear maps

Suppose $A = (a_{ij})$, $1 \leq i \leq m$, $1 \leq j \leq n$ is an $m \times n$ matrix. then $A: \mathbb{R}^n \to \mathbb{R}^m$ given by $$ A(x) = A(x_1,...,x_n) = ( \sum_{j=1}^n a_{1j}x_j, ..., ...
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1answer
108 views

Homeomorphism with a bouquet of two circles

I understand why removing two points from $\mathbb R^2$ gives a surface that is homeomorphic to a bouquet of two circles. But can someone please write this homeomorphism?
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3answers
798 views

Inverse image of a compact set is compact

Let $X$ and $Y$ be topological spaces, $X$ compact, $f : X \to Y$ continuous. Then the preimage of each compact subset of $Y$ is compact. With the stipulation that $X$ and $Y$ are metric spaces, this ...
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1answer
48 views

Does every free filter contain the cofinite filter?

In the answer to this question a free ultrafilter is shown to contain the cofinite filter. But does every free filter contain it too? Obviously the ultrafilter that extend it does.
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2answers
82 views

The structure of $\mathbb{R}^n$ [closed]

$\mathbb{R}^n$ has very good topological property, it is locally compact,etc. However, I couldn't figure out why it has such good property aside from other topological space. Can anyone answer my ...
2
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1answer
58 views

Does this claim depend on topology?

An open rectangle is a set $R\subset\mathbb{R}^2$ of the form $R=\{(x,y)\in\mathbb{R}^2;\;a<x<b\text{ and }c<y<d\}$, where $a,b,c,d\in\mathbb{R}$, $a<b$ and $c<d$. Let $\|\cdot\|$ ...
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9answers
936 views

Why did we define the concept of continuity originally, and why it is defined the way it is?

The concept of continuity is a very important idea in topology. Though I am using it all the time, but indeed I don't know what is the original purpose for us to define this concept. And I also don't ...
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1answer
61 views

Metrizable spaces

A topological space X is metrizable if it is homeomorfic to a metric space. I want to know does this mean that all of topological properties of a metric space inherit to that topological spaces? Also ...
3
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1answer
194 views

A subset of $\Bbb R^p$ is open iff it is the union of a countable collection of open balls

I am studying analysis on my own and need some help verifying the solution to the above exercise found in Bartle's Elements of Real Analysis. I know there are other posts answering the same question ...
2
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1answer
126 views

Brouwer's fixed point theorem

Theorem: If $f:D^n\rightarrow D^n$ is continuous then there is $x \in D^n$ such that $f(x)=x$. To prove the theorem we assume that $f$ is cts but has no fixed point, that is $f(x)\neq x$ for all ...
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1answer
49 views

Upper half plane with irrational slope topology.

Let $X$ be the upper half plane $\{(p,q);p,q \in \mathbb Q, q\geq 0 \}$ with the Irrational slope topology. Can we say that the set $$\{(p,q);p,q \in \mathbb Q, q> 0 \}$$ is a closed set of $X$. I ...
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1answer
32 views

The upper Fell topology on Hyperspaces

Let $2^X$ be the set of closed subsets of $X$. For every subset $A \subset X$, we define: $A^+=\{ F \in 2^X ; F \subset A \}$. Let, $\Delta$, be the set of finite subsets of $X$. Then the upper fell ...
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2answers
96 views

Not well-ordered sets satisfying the least upper bound property

Before proving that every closed interval in $\mathbb{R}$ is compact, James R. Munkres remarks that, in Section 27 entitled "Compact Subspaces of the Real Line" of Topology (2nd edition), Remark: ...
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1answer
204 views

Do the supremum and infimum always exist for convergent sequences

If $\{x_n\}$ is a real sequence converging to $x \in \mathbb{R}$, do $\displaystyle \sup_n{x_n}$ and $\displaystyle\inf_n{x_n}$ exist? I think yes as, choosing $\epsilon = 1$ we have for some ...
3
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2answers
75 views

If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$

I have a proof for this question, but I want to check if I'm right and if I'm wrong, what I am missing. Definitions you need to know to answers this question: $\epsilon$-neighborhood, interior points ...
2
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1answer
71 views

Open maps and product topologies

This is my problem: Let $X$ and $Y$ be topological spaces with topologies $T_X$ and $T_Y$, respectively. Recall, a map $X \xrightarrow{f} Y$ is called open if $\forall U \in T_X, f(U) \in T_Y$. ...
2
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1answer
97 views

Fixed point theorem on a compact set

LEt $X\subseteq \mathbb{R}^d$ be a closed set, and $f: X \to X$ be a function such that $$ ||f(x_1) - f(x_2)|| < ||x_1 - x_2|| \; \; \forall x_1, x_2 \in X $$ If $X$ is compact, then there exists ...
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3answers
99 views

In a topology, if a third open set is formed of intersection of two open sets. How is it Hausdorff?

A very basic question. If a Topology, $T$ has open sets $O_1$ and $O_2$, and $O_3$ is their intersection or union then how can the neighborhoods of $O_3$ and $O_2$ or $O_3$ and $O_1$ be disjoint, ...
2
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1answer
291 views

Proving a topological space is/is not Hausdorff?

I know this is a basic question, but I am having trouble proving that particular topological spaces are/are not Hausdorff and was wondering if I could get some guidance. For example, I have to decide ...
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0answers
36 views

Completeness and being totally bounded

Are completeness and being totally bounded somehow related with each other for uniform spaces?
3
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1answer
93 views

US does not imply AB

We say that a topological space $X$ is: $AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A ...
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3answers
451 views

Prove the int(int(S)) = int(S)

I had a question on what exactly I need to show in proving that the interior of the interior of a set is equal to the interior of a set. I'm given a set $A\subset X$, where $X$ is a topological ...
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2answers
77 views

Euler characteristic of a Y-shaped pipe?

I'm familiar with the idea in topology that shapes that can be continuously deformed into one another are considered "equivalent". I read about the Euler Characteristic as being Vertices-Edges+Faces. ...
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3answers
178 views

Clarification on trivial topology not being metrizable.

This is my understanding of the proof. Please let me know if I am correct. If I take any two points in a trivial topology. I can find the minimum radius such that they both are in two different open ...
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1answer
248 views

continuous real valued functions on the ordinal space $[0,\Omega)$

As a continuation to this question. Let $X$ be the ordinal space $[0,\Omega)$, with the order topology, where $\Omega$ is the first uncountable ordinal. Let $f:X \rightarrow \mathbb R$ be a ...
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1answer
16 views

every single element of a topology T is closed if and only cofinite topology is contained in T

I 've been told that every single point of a topology T is closed if and only cofinite topology is contained in T. I am struggling to prove this. I was thinking that indeed if every single element is ...
2
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1answer
39 views

If $X$ is not countably compact, then there exists a countable subset without accumulation points

I want to prove that if $X$ is not countably compact, then there exists a countable subset $\{x_n:n\in\mathbb{N}\}$ and has no accumulation points. If $X$ is not CC, then there exists an open cover ...
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2answers
119 views

Why topology is called Rubbersheet Geometry?

Usually topology classes starts with comparing doughnut and tea cup. But after introductory class teacher will move to the definition of topology as a collection of subsets of a set having certain ...
3
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1answer
417 views

Simplicial Complexes, Triangulation general question.

I am taking a first course in topology, and I am struggling with simplicial complexes. Specifically the triangulation of subspaces of $ \mathbb{R}^n $ confuses me. If you could help me on the ...
2
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1answer
59 views

Intuition for an open mapping

What is an intuitive picture of an open mapping? The definition of an open mapping (a function which maps open sets to open sets) is simple sounding, but it's really not as easy to picture as the ...
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2answers
1k views

Difference between closed, bounded and compact sets

In real analysis, there is a theorem that a bounded sequence has a convergent subsequence. Also, the limit lies in the same set as the elements of the sequence, if the set is closed. Then when ...