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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67 views

Show that a set is closed and open

We say that $f: X \to M$ is bounded if $f(X) \subset B_r(a) = \{m \in M \mid d(m,a) \lt r\}$ for some $r \in \mathbb{R}^+$ and $a \in M$. (NOTE: $(M,d)$ is a metric space with a distance $d$). We ...
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3answers
42 views

Showing that $F$ is closed, given that it contains all of its limit points

Let $X$ be a Hausdorff topological space. I am trying to show that if $F \subset X$ has the property that if $\{x_n\}_{n \in \mathbb{N}}$ is a sequence in $F$ that converges to $x_0 \in X$, we must ...
2
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3answers
53 views

the rationals as a metric space, closed and open sets

If we consider the rationals, $\mathbb{Q}$, as a metric space with the usual metric of the real line, then is the set $B=\{q\in\mathbb{Q} \mid {q^2}\lt 2\}$ a closed set? I know the definition of a ...
2
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1answer
42 views

What, exactly, is a vertical homotopy?

As the question title suggests, what exactly is a vertical homotopy? Googling has failed to provide any results as so far as a clear definition goes...
2
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1answer
23 views

Quotient topology of a topological vector space is translation-invariant

Let $(L,\tau)$ be a topological vector space over $\Bbb{C}$ and $M$ be a subspace of $L$ and let $$f:L\to L/M$$ be the canonical map of $L$ onto $L/M$.Let $ \hat \tau$ be the quotient topology on ...
2
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1answer
30 views

Basic topology proof regarding injection being equal to intersection property for two subsets

Let $f: X \rightarrow Y$ be a map from $X$ to $Y$. Show that the statement $$f(C \cap D) = f(C) \cap f(D)$$ for all possible choices of $C, D, \subseteq X$ is equivalent to $f$ being injective. So ...
2
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4answers
66 views

Show that $E_r(a)=\{x\in\mathbb{R}^n:\|x-a\|>r\}$ is path-connected

For $a\in\mathbb{R}^n, r>0,n\ge2,$ show that $E_r(a)=\{x\in\mathbb{R}^n:\|x-a\|>r\}$ is path-connected and hence connected. So I'm trying to use the idea that for x, y $\in E_r(a)$ if ...
2
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2answers
37 views

Example 13, Sec. 3 in Munkres' TOPOLOGY, 2nd ed: Necessary and Sufficient Conditions for a Subset to Have the Least Upper Bound Property

Let $A$ be an ordered set having the least upper bound property, and let $A_0$ be a non-empty subset of $A$. Then what is (are) the necessary and sufficient condition(s), if any, for $A_0$ to have ...
2
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1answer
53 views

Archimedean set

right now i'm taking a topology class, where the first chapter is named :"real numbers and sequences": (in french it's called : Nombres et suites réels) , i was wondering if anyone here could clarify ...
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2answers
116 views

Prove that open half planes are open sets

An open half plane is a subset of $\Bbb{R}^2$ in the form $\{(x, y)\in \Bbb{R}^2\vert \space Ax + By<C\}$ for some $A,C,B\in \Bbb{R}$ with either $A$ or $B$ nonzero. I need to prove that open ...
2
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1answer
95 views

$G$-sets, natural correspondence?

See here. In the category of $G$-sets, the morphisms $f:G/H\to X$ are in one-to-one correspondence with the elements of $X^H$; the correspondence sends $f$ to $f(H)$ (where the subgroup $H$, being ...
2
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1answer
35 views

Question on quotient space

I was recently given this product space $X\times [0,1]$ with equivalence relation generated by $(x, 1) ~(y, 1)$. How to show the quotient space is connected? I just cannot do it Thanks all.
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1answer
49 views

Proving set is open using continuous function

Let S = $ \{(x,y) \in \mathbb{R}^{2}| x^{2} + y^{2} =1 \} $ I want to prove that this set is closed or by showing that it's complement is open. Now my proof becomes show that $S^{c}$ = $ \{(x,y) \in ...
2
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1answer
68 views

Is There a Generalization of the Path Lifting Property Of Covering Maps.

$\newcommand{\R}{\mathbf R}$ Let $p:(E, e)\to (X, x)$ be a covering projection map. We know that for any path $\gamma:I\to X$ such that $\gamma(0)=x$, there is a unique lift $\Gamma:I\to E$ such that ...
2
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1answer
95 views

On the terms “nowhere dense” and “dense-in-itself”

Are there literal interpretations of the terms "nowhere dense" and "dense-in-itself" from which these terms' definitions follow? If I were to guess what it means for a subset $A$ of a topological ...
2
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1answer
24 views

Forming the orthogonal space is just a special case of forming a polar set?

Let $M\subset X$ be a subspace and define the polar set $M^{\circ}:=\{x^{\ast}\in X^{\ast}:|\langle u,x^{\ast}\rangle|\le 1\forall u\in M\}$ and $M^{\perp}:=\{x^{\ast}\in X^{\ast}:\langle ...
2
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1answer
47 views

Longest sequence of minimally finer topologies part 2

Suppose we start with a topology $T_1$ of X. Is there a way to get construct a sequence of topologies $T_n$ such that $T_{n - 1} \subset T_{n}$ in which there is no finer topologies in between, also ...
2
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1answer
34 views

Continuous function of the Hyperspace of Compact Sets

Let $X$ be separable and completely metrizable. Let the hyperspace of compact sets be denoted $H(X)$. I want to show that $H(X)^2 \rightarrow H(X)$ defined by $(H,G)\rightarrow H \cup G$ is ...
2
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2answers
92 views

How to show that $S_\Omega$ satisfies the sequence lemma?

I am trying to prove this fact, but there is one thing that shows up that makes me unable to finish the proof, do you have any tips on how to finish the proof? A space X satisfies the sequence lemma ...
2
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2answers
54 views

Topologically equivalent metrics? Ceiling function of metric $d$

I am asked if the following metrics are topologically equivalent or not. $(X,d)$ is a metric space and $d$ is the metric. Define $\lceil{d} \rceil (x,y)$ := $\lceil{d(x,y)} \rceil$:$X \times X ...
2
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1answer
51 views

Boundary of set on relative topology in $R^n$

Let A be a subset of $R^n$ with the relative topology, let B be a subset of A, is the boundary of B in the relative topology on A equal to the intersection of A with the boundary in $R^n$? and ...
2
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1answer
49 views

Heine-Borel in $\mathbb{R}^2$

I want to prove this variation of Heine-Borel theorem in $\mathbb{R}^2$ in the following way. Theorem. The square $C=[c_1-r,c_1+r]\times[c_2-r,c_2+r]$ is compact. Here is the idea for the proof. ...
2
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2answers
46 views

Prove that $\mathbb{R}^n$ without the origin is an open set with the euclidean metric?

I was thinking that i could prove that all elements in $\mathbb R^n-(0,0,...,0)$ are interior points. That meaning if given any $p\in \mathbb R^n$ there exists $\epsilon>0$ such that ...
2
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2answers
63 views

Need a counterexample related to Interior and Closure [duplicate]

Let "cl" denotes closure, "int" denotes interior.I'm looking for a Single example of a subset $A$ of some topological space $X$ where all the following sets are unequal: $1.$A $2.$ int(A), ...
2
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1answer
122 views

Equivalence of two definition of compactness.

Definition 1 : The set $A\subset \mathbb R$ is compact if it's closed and bounded. Definition 2 : The set $A\subset \mathbb R$ is compact if for all cover $\mathcal U$ of open set there is ...
2
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3answers
36 views

About diameter in a metric space.

$\newcommand{\diam}{\operatorname{diam}}\newcommand{\cl}{\operatorname{cl}}$Let $(X,d)$ a metric space and $A \subseteq X$. Then $\diam(A)=\diam(\cl(A))$ I could prove $\diam(A) \leq \diam(\cl(A))$. ...
2
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1answer
51 views

What is entailed by a homeomorphism with a discrete metric?

I have a homework question in which I am asked to prove that several statements are equivalent. I'm confused about a particular step in this chain, namely $a)\implies b)$ below. Given: Let $M$ be ...
2
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1answer
71 views

What is the name of this solid?

You have a sphere. Take it and drill a hole along a diameter. You have a torus. Then rotate the sphere 90 degrees and drill along another diameter. There are now two perpendicular, intersecting ...
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2answers
54 views

Definition of Topology on $\mathbb{R}$ via Neighbourhoods

A topology can be defined via different axioms. One of them see here, in terms of neighbourhoods, states a topology is a set $X$ endowed with neighbourhood function $N:X\to\mathcal{F}(X)$, where ...
2
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1answer
70 views

Proof of Darboux Theorem in topology

$I$ is an open interval in $\mathbb{R}$, $f:I\rightarrow \mathbb{R}$ is a differentiable function. $T=\{(x,y)\in I\times I:x<y\}$. Let $g : T → \mathbb{R}$ be the function defined by $g(x, y) = ...
2
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2answers
92 views

Neighborhood base at zero in a topological vector space

I'm reading a proof of a theorem which characterizes the collections of sets which can serve as a neighborhood base at zero in a topological vector space: Here are my questions: Why "This shows ...
2
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1answer
47 views

Find the connected components of $X=\{(x,y)\in\mathbb{R}^2 :x\neq y\}$ with the topology induced from $\mathbb{R}^2$.

My work: $\{(x,y)\in\mathbb{R}^2 :x< y\}$ is path-connected, and hence is connected. It's also maximal, as if we were to add a point $(x,y)$ where $x=y$, then it wouldn't belong to $X$, and if we ...
2
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1answer
126 views

Show that the product topology on $X \times Y$ is the same as the metric topology, where the metric is the product metric

Any help on the following problem would be greatly appreciated. Thanks! Given metrics $d$ and $e$ on sets $X$ and $Y$, let $f$ be the product metric on $X \times Y$. So ...
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1answer
30 views

Let $f: A \times B \rightarrow C$ be continuous and closed under product of closed subsets of A and B, is $f$ closed?

Assume product topology on $A \times B$. To make clear th title: $f$ is a countinuous map such that if $R \subset A$ and $S \subset B$ are closed sets, then $f(R \times S) \subset C$ is a closed set ...
2
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1answer
69 views

Orbit of shift operator in symbolic dynamics

Consider the symbolic space of infinite $0-1$ words, $\{0,1\}^{\mathbb{N}}$. This can be shown to be a compact metric space with $$d(s, \bar{s}):=\displaystyle\sum_{t = ...
2
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1answer
58 views

$d(K,\partial U)=\inf\{d(K,\partial U_{\alpha}):\alpha\in A\}$

Let $\{U_{\alpha}\}_{\alpha\in A}$ be a family of open sets in $\mathbb{R}^n$. Let $U$ be a connected component of $$\mathrm{int}\left(\bigcap_{\lambda\in A}U_{\alpha}\right)$$ and $K\subset U$ a ...
2
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1answer
46 views

Int M is open and a manifold

If M is an n-dimensional manifold with boundary, then Int M is an open subset of M , which is itself an n-dimensional manifold without boundary. I am supposed to use these definitions: If M is an ...
2
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1answer
74 views

Why is the measure of a boundary of an open ball positive in only a countable number of cases?

Let $X$ be a Polish (complete separable metric) space and $\mathbb{P}$ a Borel probability measure on $X$. Let $x_1, x_2, \ldots$ be a sequence of points dense in $X$. How can you prove that there is ...
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3answers
80 views

Is the intersection of the following closed and open set closed? Generally?

Ok, I have been informed that the below lemma is incorrect. I needed it to prove the following statement. Could someone else provide a proof? Statement: If m(E) is finite, there exists a compact set ...
2
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1answer
61 views

subspace of a metric space

Let $(S,d)$ be a metric space, $\mathcal{S}$ the induced topology. $A\subset S$ a subset. It is easy to see that $A\cap\mathcal{S}=\mathcal{A}$, i.e., the topological subspace on $A$ is the ...
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70 views

Why is uncountable union of $\mathbb{R}$ the same as this space

Can anyone give an intuitive reasoning as to why the uncountable disjoint union of copies of $\mathbb{R}$ is the same as $\mathbb{R}$ with discrete topology product with $\mathbb{R}$ with the usual ...
2
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1answer
59 views

How to prove that space is not connected

I found a definition that the space $M$ is not connected if there are open subsets $A,B$ such that $M=A\cup M,A\ne\emptyset\ne B,$ and $A\cap B=\emptyset$. How can I prove from the definition that ...
2
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1answer
31 views

How to check F:AxI->B is continuous

A and B are topological spaces.Let f and f' are continuous maps from A to B and homotopic.Then we need F:AxI->B,continuous,where F(s,0)=f(s) and F(s,1)=f'(s). Now my question is if we want to ...
2
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3answers
61 views

When the set of $r$-far interior points from a set is open

Let $E$ be a subset of a metric space $X$ and for $r > 0$ let $$ E_r = \lbrace x \in E : d(x,E^c) > r \rbrace .$$ Is the set $E_r$ always open? Equivalently, is the function $ x \mapsto ...
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2answers
115 views

An intitutive solution to problems relating to closed sets in topology

The question given in my homework problem is, Let $ \{A_{\alpha}\}_{\alpha \in \Lambda} $ be a family of closed subsets in an arbitary topological space $X$ . Assume that for each $x$ there exists an ...
2
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1answer
106 views

The empty set is a neighborhood?

The following axioms of a Topological space is from Wikipedia: Neighbourhoods definition This axiomatization is due to Felix Hausdorff. Let $X$ be a set; the elements of $X$ are usually ...
2
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1answer
64 views

Confusion regarding the $\omega$-limit of a set in a flow

In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot ...
2
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1answer
66 views

Open cover of non-compact spaces

Let $X$ be a non-compact space. (A space is compact if any open cover has a finite subcover.) I want to show that there is an ordinal $\alpha$ and an open cover $(U_\xi)_{\xi < \alpha}$ such that ...
2
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2answers
88 views

Algebra formed by operators and Kuratowski's theorem

I have been reading the paper "D. Sherman, Variations on Kuratowski's 14-set theorem, Amer. Math. Monthly 117 (2010), no. 2, 113-123" recently. Kuratowski Closure complement theorem states that: Let ...
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2answers
80 views

Understanding the proof of “connected set is interval.”

There is some questions about connected set. The first question arises in logical translation. I translated the property of interval into logical proposition that $\forall a,b \in E, \forall c : ...