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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Show that there exists $\epsilon >0$ such that $\bigcup_{x\in A}B(x;\epsilon)\subset V.$

Let $X$ be a compact metric space, $A$ a closed subset of $X$ and $V$ an open subset of $X$. Suppose $A\subset V$. Show that there exists $\epsilon >0$ such that $$\bigcup_{x\in A}B(x;\epsilon)\...
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1answer
156 views

Let X be a subspace of $\mathbb R^2$ consisting of points that at least one is rational. Prove that X is path-connected. [duplicate]

Let X be a subspace of $\mathbb R^2$ consisting of points such that at least one of coordinates x and y is rational. Prove that X is path-connected. A sketch is as follows. Is it right? Also How to ...
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1answer
38 views

Connectedness of sets

Let $E \subset S$. Suppose $E$ is not connected. Then in the induced topology of $E$ relative to $S$, $E$ and $\emptyset$ are not the only clopen sets. I can show using the above definition that if $...
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1answer
104 views

$S^1 \times S^1$ is homeomorphic to torus

The question is asking to show $S^1 \times S^1$ is homeomorphic to a torus. I have read some other posts in here but most of them are proving it with "lattice" which I haven't learn. Here is what I ...
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1answer
31 views

$T_{0}-$ and $T_{1}$ axioms.

Good day! I tried to prove some propositions from the separation axioms and I would just like to share it and ask if there are some missing points in my arguments or how can I improve my proof. Thank ...
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2answers
24 views

Connected subspace of the product of topological spaces

Let $X$ and $Y$ be topological spaces. Is it true that any connected set of $X \times Y$ is of the form $A \times B$ where $A$ is connected subset of $X$, and $B$ is a connected subset of $Y$?
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43 views

Continuity from property of constriction images of spheres

Let $D\subset\mathbb R^n$ --- domain and mapping $\varphi:D\to \mathbb R^n$. The following property holds There is a set $T\subset D$ s. t. measure $|D\setminus T|=0$ and for every point $x\...
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40 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
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233 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
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1answer
117 views

Segment ordered density conjecture.

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset \overline{...
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1answer
96 views

Is a diffeomorphism's image automatically open?

Sorry if this question is trivial, I am new to smooth manifold theory. Let $\varphi : I \times \mathcal S^{n-1} \to X$ be a diffeomorphism. $I=(0,1)$, $\mathcal S^{n-1}$ is the unit sphere in $\...
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1answer
56 views

topologies of real projective plane models

Consider a sphere upon a plane (say $\mathbb{R}^2$). Let $C$ be the center of the sphere. Consider the lower hemisphere plus the boundary (bowl) and project lines from $C$ across the surface until ...
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2answers
47 views

continuity of a map on a $T_2$ space

Let $X$ be a $T_2$ space .Let $f:X\rightarrow \mathbb R$ be such that $\{(x,f(x):x\in X\}$ is compact.Show that $f$ is continuous My attempt: Let $x_n$ be a sequence in $X$ converging to $x$.To ...
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309 views

Extending a continuous function to the closure

I'm dealing with the following problem: Let $X$ a topological space, $Y$ a metric space and $A$ a subspace of $X$. If $f$ is a continuous mapping of $A$ into $Y$, show that $f$ can be extended in ...
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1answer
36 views

Covering associated to a map

I'm stuck with this exercise. Let $$p : E \to X $$ be a covering map. Y is a connected and locally path-connected topological space, $$f : Y \to X $$ is a continous map. The claim is that $$f^*p :...
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1answer
76 views

Open and bounded set with compact boundary

Why does an open and bounded set in an infinite dimensional space have to be the emty set, if it has a compact boundary? And the space has a norm, by the way. Cheers, Richard
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124 views

Elementary question about topology and metric spaces

Let $(X, \rho ) $ be a metric space. Denote $$ \mathscr{B} = \{ B(\epsilon,x) : x \in X, \epsilon>0 \} $$ Let $B_1,B_2 \in \mathscr{B}$ . Let $x \in X$ be arbitrary with $ x \in B_1 \cap B_2 $. ...
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1answer
190 views

Compact subspace of $\mathbb{R}$ with lower limit topology must be countable.

Any compact subset of $\mathbb{R}_{l} $ must be a countable set. Consider the open cover $\{[n,n+1): n \in \Bbb Z\}$ of $\Bbb R $ which has no subcover. So $\Bbb R $ is not compact with respect to ...
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3answers
67 views

If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact.

If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact. Disproof by counterxample? Not true. Let $X = \mathbb{R}$ with the usual topology and $A = (-\infty,0)$. ...
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1answer
67 views

Is Every Set Open in the Subspace Topology on the Cantor Set?

Im working in the real line with the usual topology. For the cantor set subspace of R, let T represent the the subspace topology on the cantor set induced by the usual topology. I'm trying to show ...
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1answer
97 views

Show that ${\mathscr C}(\{1,..,n\},R)$ and $R^n$ have the same open sets

Question: Let X be the set $\{1,2,...,n\}$ equipped with the discrete metric ($\delta(x,y)=0$ if $x=y$, $\delta(x,y)=1$ if $x\neq y$). Then ${\mathscr C}(X, R)$ and $R^n$, where $R$ is the real ...
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1answer
48 views

Problem of a compact space

Let $X$ be a compact $T_2$ space.Assume that the vector space of real-valued continuous functions on $X$ is finite dimensional.Show that $X$ is finite. Spent nearly 3 hours on this problem.Cant ...
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1answer
93 views

Continuous function that has limit at infinity is uniformly continuous (another viewpoint)

I know how to prove that, given a continuous $f:[0,\infty) \rightarrow \mathbb{R}$ such that $\displaystyle \lim _ {x \rightarrow \infty} f(x)=L$, then $f$ is uniformly continuous (by means of taking ...
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1answer
101 views

$[0,5]$ is not compact with the Sorgenfrey topology

Show that in the Sorgenfrey topology $[0,5]$ is not compact. Justify your answer. Here is my shot at an answer. Could anyone please knock it down/improve it/help with the correct answer? ...
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3answers
110 views

In $\mathbb R$ with the usual topology, prove that the set of rationals is not compact.

In $\mathbb R$ with the usual topology, prove the set of rationals is not compact. Here is my attempt at a proof by contradiction. If $\mathbb Q$ is compact, then $\forall \{G_{\alpha}\}_{\alpha}$ ...
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1answer
88 views

equivalent metrics and uniform equivalent metrics

Let (X,d) be the Euclidean metric on the real number, and define σ(x,y)=min{1,d(x,y)} if if x, y are rationals or x, y are irrationals, and σ(x,y)=1 otherwise. I would like to study if these metric ...
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3answers
56 views

Interior of a set?

I'm trying to think if their is any topology for which this is false: If G is an open set, then G = interior(G) Can anybody think of anything? I'm pretty sure it's straight forward.
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86 views

Book on “topology ” for starters [duplicate]

This semester I have a course on topology. I'd like you to recommend me some books.
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1answer
80 views

Proving the set of all real matrix $3\times2$ with rank $2$ is an open subset of $\mathbb{R}^{3\times2}$

I have to prove that the set of all real matrix $3\times2$ with rank $2$, M, is an open subset of $\mathbb{R}^{3\times2}$. I do not know how to do it, but trying to prove that $\mathbb{R}^{3\times2}-M$...
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0answers
102 views

highway metric topologically equivalent to euclidean metric?

Consider the Euclidean metric space $(S, d_1)$ on $\mathbb{R^2}$ and the highway metric space $(S, d_h)$ on $\mathbb{R^2}$, where the highway metric is defined as $$d_h(x,y) = \begin{cases} |x_2 - ...
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2answers
100 views

equivalent metric space

Let $(X, d)$ be a metric space where $d$ is unbounded, that is, $$\sup\{d(x; y) : x, y\in X\} = \infty$$ Define a bounded metric $p$ on $X$ such that: $(i).$ $f : (X, d) \rightarrow (X, p)$, $f(x) = ...
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1answer
70 views

Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism

It is a Dugundji's exercise: Show that if $X$ is $T_2$, $f \colon (X, \tau) \to (Y, \sigma)$ continuous in $X$ and $f|_{D}$ homeomorphism with $D$ dense in $X$ then $f(X - D) \cap f(D) = \varnothing$....
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2answers
393 views

Prove that $SL(n,R)$ is connected.

Prove that $SL(n,R)$ is connected. The problem is I know only topological groups from Munkres only. Again Just started fundamental groups. So if anyone can explain me how it is true in a lucid ...
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1answer
53 views

Convergence of filters

Let (X,T) be a topological space. If F is a filter on X, then B:={G⊆X |G∈T and G∈F} is a basis for a filter F° on X. Prove that for x ∈ X the filter F° converges to x if and only if F converges to x. ...
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1answer
57 views

Property of a compact metric space

Let $(X,d)$ be a compact metric space.Suppose that $f:X \rightarrow X$ is a function such that $d(f(x),f(y))<d(x,y); x\neq y$ Show that $f$ has a fixed point. Tried to prove this using ...
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1answer
41 views

Is the set $ \{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$ connected?

Let $X=\{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$. Is $X $ connected or disconnected? My attempt:$X$ is connected iff any two points of $X$ are contained in a connected subset of $X$. This attempt ...
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115 views

For a subfield $K$ of $\Bbb C$ with $K\nsubseteq \Bbb R$, show $K$ is dense in $\Bbb C$. [duplicate]

Let $K $ be a subfield of $\mathbb C$ not contained in $\mathbb R$. Is $K$ dense in $\mathbb C$? My problem is I have never used the concept of dense set in algebra and neither have any idea about ...
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167 views

Is there a rationality-preserving order isomorphism between $\mathbb{Q}$ and two disjoint open intervals?

I have a homework question in a intro logic course, part of which requires me to Find an order preserving isomorphism between $\mathbb{Q}$ and $\mathbb{Q} \cap ((0,1) \cup (2,3))$. So, I need an ...
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1answer
91 views

Continuous functions from a topological space to itself.

Recently I studied the theorem that states that if X is a topological space equipped with the discrete topology, then every function from X to itself must be continuous. I am wondering if the converse ...
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46 views

limit point of $\frac{1}{m}\sin(m)$ for $m \geq1$

how do I show that 0 is the only limit point of $\frac{1}{m}\sin(m)$ for $m \geq1$ integer? It is clear that 0 is a limt point since it is the limit of this sequence, but I cannot prove that there are ...
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1answer
105 views

Creating topological spaces with portals

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension. I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
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346 views

Munkres Chapter 1 Section 7 Exercise 8

Let $X$ denote the two element set $\{0,1\}$; let $X^\omega$ denote the set of all the binary sequences; and let $B$ denote the set of countable subsets of $X^\omega$. Then how to see if $X^\omega$ ...
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96 views

What is the topological properties of $\mathbb R $ that makes it uncountable (as compared to $\mathbb Q $)?

What is the topological properties of $\mathbb R $ that makes it uncountable (as compared to $\mathbb Q $)? Further, what axioms (or properties) of $\mathbb R $ do these topological properties depend ...
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2answers
50 views

How to define continuity of functions from $R$ to $P(R^2)$?

Consider a 2-dimensional amoeba that moves in $R^2$. This amoeba can be defined as a function $f$ from a real interval to $P(R^2)$: the real interval represents the time, and $P(R^2)$ (= the subsets ...
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46 views

Is a set open in a product of spaces if all its segments are open in their factors?

Let $X$ and $Y$ be topological spaces and let be $U ⊂ X × Y$ such that $$∀(x,y) ∈ X × Y \colon \quad \begin{aligned}•~~\mathrm{incl.}_{(–,y)}^{-1}(U) ~\text{is open in $X$,}\\•~~\mathrm{incl.}_{(x,–)}^...
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1answer
150 views

homeomorphism between a boundary and a sphere

I started another question related to this. Consider $|\Delta^n|:=\{(x_0,..,x_n)\in\mathbb{R}^{n+1}:\sum_{i=0}^n x_i=1, x_i\in[0,1]\;\forall i\}$ the geometric realization of the standard n-simplex $\...
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1answer
40 views

Homeomorphism of quotient spaces

If $X$ and $Y$ are homeomorphic spaces with homeomorphism $f:X\rightarrow Y$, is it true that for any subspace $A$ of $X$, $X/A$ and $Y/f(A)$ are homeomorphic? If so, how to show this?
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1answer
76 views

Show that a space X is homeomorphic to the space of multiplicative linear functionals

Let $\mathcal{A}=C(X,\mathbb{R})$ where $X$ is a compact Hausdorff space. Let $\hat{\mathcal{A}}$ be equal to the set of multiplicative linear functionals from $\mathcal{A}$ to $\mathbb{R}$. ...
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219 views

Connected sum of orientable manifolds

I was reading through Lee's Smooth Manifolds on the part regarding orientations and I was wondering if the connected sum preserves the orientability of manifolds. Intuitively it seems to be true, but ...
6
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1answer
54 views

Possible behaviours of the cofinality-of-neighbourhood function for topological spaces $X$ that are connected.

Given a topological space $X,$ there is a function $f_X : X \rightarrow \mathrm{Cardinals}$ defined as follows. Given a point $x \in X$, $f_X(x)$ is the cofinality of the poset whose elements are the ...