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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2answers
65 views

A real function on a compact set is continuous if and only if its graph is compact

I have a problem with the following Let $X$ be a compact subset of $\mathbb{R}$ and let $f$ be a real-valued function on $X$. Prove that $f$ is continuous if and only if $\{(x,f(x)) \mid x\in X\}$ ...
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0answers
18 views

interior/closure of a convex set [closed]

I have some problem with the following Let $C$ be a convex set in $\mathbb{R}^n$. (i) Assume that $C$ has nonempty interior. Let $x_1\in \text{cl}(C)$ and $x_2\in \text{int}(C)$. Show that any ...
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1answer
31 views

For $f(x, y) = x-y$, is $f(K \times K)$ closed if $K$ is closed?

$f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x, y) = x-y$. For $K \subset \mathbb{R}$ closed is $f(K\times K)$ closed? For the closed interval this is straight forwardly true ...
4
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0answers
42 views

Closed or open subsets of $C[a,b]$?

$C[a,b]$ denotes the space of continuous real-valued functions on $[a,b]$. The metric associated with $C[a,b]$ here is $d(f,g)=sup[|f(x)-g(x)|]$ where the supremum is taken over $[a,b]$. $C^1[a,b]$ ...
2
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2answers
50 views

Show that $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$ is continuous, with $f$ continuous.

The entire problem statement is, Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. Define $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$. Show that $\Gamma_f$ is continuous. My attempt ...
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2answers
36 views

Is the following proof about continuous function on open&closed intervals correct?

So I want to prove that there exist no continuous onto function $f$ from $[0,1]$ to $(0,1)$. To do so, I argue as follows: Suppose for contradiction that $f:[0,1]\to(0,1)$ is an onto continuous ...
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3answers
62 views

Is a path connected covering space of a path connected space always surjective?

If $X$ is a path connected topological space, a covering space of $X$ is a space $\tilde{X}$ and a map $p:\tilde{X} \to X$ such that there exists an open cover $\left\{ U_\alpha \right\}$ of $X$ where ...
1
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3answers
54 views

Show that $\mathbb{R}^k$ is separable

A metric space is called separable if it contains a countable dense subset. I have no idea how to go about proving this. What sort of things should I understand to do this. What would a sample proof ...
5
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2answers
76 views

one-point compactification of a compact space

We have the definition of the one-point compactification of a locally compact Hausdorff space: Let X be locally compact and Hausdorff and let Y be a compact Hausdorff space and $i:X\to Y$ such that ...
2
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1answer
65 views

How can I visualize principal bundles?

I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization $ϕ^{-1}_i:π(U_i)→U_i×G$ . For example $ϕ^{-1}_i:π(S^2)→S^2×U(1)$ (if that makes sense) ...
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0answers
19 views

Continuous images of convergent sequences in metrizable spaces are convergent [closed]

A) Show that a map F from a metrizable space X is topological space Y is continuous if and only if for every convergent sequence{xn}in X converging to x,the sequence {f(xn)}converges to f(x) ...
2
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0answers
26 views

Complement of contractible subset of a sphere

Let $A$ be a nice closed subset of the sphere $S^n$; for example, we could ask $A\to S^n$ to be a cofibration. Assume that $A$ is contractible. Is then $S^n - A$ also contractible? It ...
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0answers
39 views

General definition of piecewise continuity

Is there a general definition of piecewise continuity for functions between topological spaces ? Of course one can intuitively says that $f: X \rightarrow Y$ is piecewise continuous if for every ...
2
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2answers
40 views

Is there example of two disjoint closed sets in Rational Sequence topology.

The set of the real numbers with the Rational Sequence Topology is not normal by Jone's Lemma. What is a sample of two disjoint closed sets that cannot be separated by two disjoint open sets?
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1answer
35 views

Automorphism of fundamental group of torus

I am asked to show that every automorphism of the fundamental group of a torus $T=S^1\times{}S^1$ is induced by a homeomorphism $h:T\rightarrow{}T$, which fixes the base point. What I was thinking is ...
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1answer
101 views

What are all kind of “metamath” good for? Can it help me here?

Those logical theories, which deals with questions that isn't really mathematics but reach mathematics more or less, often seems to be like textbooks full of definitions, plus some theorems of the ...
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1answer
30 views

Small question about the existence of an homeomorphism

I need to prove this small fact to understand the proof of a theorem: Suppose $X, Y$ are countable discrete subespaces of $\beta \omega$, $Y \subset X$, and let $p \in \beta \omega \setminus \omega$ ...
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1answer
48 views

Order-preserving embeddings

(Follow-up to Existence of a utility function on the reals.) Say we have a totally ordered set $X$ which has a countable, dense subset $C$. I believe we can find an $f:C\to\mathbb R$ which is ...
0
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1answer
28 views

Is $\mathbb{R}$ is the union of a negligible and a meager set?

A subset of $\mathbb{R}$ is said to be "negligible" iff it is of Lebesgue measure zero, and "meager" iff it is contained in a countable union of nowhere dense closed sets (i.e., closed sets with empty ...
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0answers
23 views

K topology interpretation

I just want to make sure i am interpreting the k topology correctly. My concern is with the negative values. Is a basis element such as (-1, -1/2) not in the k-topology? But it states that all ...
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3answers
27 views

Prove that the closure of complement, is the complement of the interior

Let $(X,d)$ a metric space and $A\subset X$. Prove that $$\overline{(A^c)}=\overset{\circ}{(A)}^c$$ i.e. the closure of complement, is the complement of the interior. Proof. If $x\in ...
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1answer
46 views

$\phi : S^n \to S^n$ with no fixed point

The question is as follows: "Find a continuous map from $S^1$ to $S^1$ with no fixed points. What about for $n > 1$?" I want to write $S^n = \{(1, \theta_1, \dots, \theta_n) | 0 \leq \theta_i < ...
4
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2answers
35 views

Topological Group Structure on a Subset of $\mathbb R$

I have been pondering over this question for quite a long time. Please help. Is it possible to define some group operation on $[a,b]$ (with the usual topology inherited from $\mathbb R$) so that it ...
0
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1answer
28 views

A question about dimension and connectedness in order topologies

Does there exist a totally disconnected topological space T whose topology is the order topology of a linearly ordered set and whose (small inductive) dimension is equal to 1? There exist topological ...
0
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1answer
20 views

Basis Definition clarity

For $\mathcal{B}$ to be a basis of some topology $\mathcal{T}$ on $X$, my book says: 1) For each $x \in X$, there is at least one basis element $B \in \mathcal{B}$ containing $x$. 2) If $x \in B_1 ...
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1answer
25 views

Open sets in normal spaces

My question is related to a doubt about a proof of Urysohn's lemma. Suppose that $X$ is a normal, $T_1$-space. Can we find a basis for $X$ consisting of sets $U$ such that $V\subseteq ...
2
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1answer
112 views

How do I prove this function is not continuous?

Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$. The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map ...
3
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3answers
43 views

Set $S$ which is path-connected, but $\overline{S}$ is not path-connected

Suppose the set $$S := \left\lbrace x+i \sin \left( \frac{1}{x} \right) \Bigg\vert x \in (0,1]\right\rbrace \subseteq \mathbb{C}. $$ I want to show that $S$ is path-connected but $\overline{S}$ is ...
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0answers
33 views

strengthening Lesbegue Number Lemma

Let $F : I \times I \rightarrow X$ be a continuous map and U, V be two open covers of X. Then Lesbegue lemma says that there exist partitions of I which are $0=s(0)<...<1=s(m)$ and ...
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1answer
49 views

Hausdorff property of $\mathbb{RP}^n$ from unusual definition

Rather than defining the topology on $\mathbb{RP}^n$ as the quotient $(\mathbb{R}^{n+1}\backslash\{0\})/$~ or $S^n/$~ in the usual way, suppose you use these equivalence relations simply to define a ...
2
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0answers
32 views

Isotropy subgroup is closed

I am studying a book and I am asked to prove the following: Show that the isotropy subgroup (for a certain right action of a topological group $G$) of $y\in Y$ is closed for $Y$ some topological ...
3
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5answers
65 views

Show that $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=x+y$ is continuous using open sets.

The problem statement is, Show that $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=x+y$ is continuous using open sets. I know that to show $f$ to be continuous I take an arbitrary open set ...
1
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2answers
79 views

Abstracted Metric and Measure Spaces

As I am just beginning to study general topology and metric spaces in more and more detail, it seems to me that the metric space topology is entirely determined by the properties of $\Bbb R$, since ...
0
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0answers
24 views

Set of semi infinite intervals on the real line is a topological basis

Show that the set $\{(r,\infty): r \in \mathbb{R}\}$ is a basis for a topology on the set of real numbers but not a topology itself. Any feedback on my proof would be appreciated. A collection of ...
4
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1answer
280 views

Klein bottle homeomorphic to union of Möbius strip

I'm having trouble showing that the Klein bottle defined as a quotient space of $I^2$ with relation $(x,-1)R(x,1)$ and $(-1,y)R(1,-y)$ is Hausdorff and that it can be expressed as $X\cup Y$ where ...
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0answers
52 views

Klein bottle visualization, parameterization, and isotropic version

Suppose a bright glowing orange mobius strip appeared in space for just an instant and then disappeared, except for its glowing orange edge, which remains suspended motionless in space for a moment, ...
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1answer
79 views

Is the following subset of a plane connected? (picture)

It's the union of a sequence of interlocked "chains" formed from closed semicircles. The chains can be seen having different shades of gray in the picture. There's no line in the middle, just the ...
3
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2answers
80 views

Is the Projective Real Plane Compact?

I feel like $\Bbb P (\Bbb R^2)$ is compact, but I know that $\Bbb R^2$ is locally compact, therefore it has a one-point compactification. $\Bbb P (\Bbb R^2)$ adds more than one point to the real ...
3
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2answers
56 views

Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the product topology on $\mathbb{R}^m \times \mathbb{R}^n$

I'm attempting to teach myself topology for graduate school this summer, but I'm having a tough time. I'm trying to prove that the Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the ...
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5answers
57 views

Non-Metrizable Topological Spaces

What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. ...
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1answer
23 views

Net based on finer filter is a subnet? [duplicate]

Let $G$ be a filter finer than the filter $F$. Is it necessary that the net based on the filter $G$ is a subnet of the net based on $F$?
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4answers
86 views

Prove that $\text{int(intA)=int(A)}$?

I want to prove that $\text{int(intA)=int(A)}$ (and we are in metric space). I have two questions regarding this. (1). I came up with this proof but don't know if it's correct or not. First I use ...
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3answers
27 views

Open Cover of Compact Set Minus a Point on the Boundary

I am having a hard time thinking of an infinite (uncountable or not) open cover of a compact set missing a point on its boundary in $\Bbb R^2$, so that the open cover has no finite subcover. I know ...
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3answers
447 views

Why are the empty set and the set of all real numbers both open and closed?

sorry! am not clear with these questions why an empty set is open as well as closed? why the set of all real numbers is open as well as closed?
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3answers
54 views

Boundary points of a domain bounded by a continuous curve

Suppose $F:\mathbb{R^2}\to \mathbb{R}$, which is given by $F(y_1,y_2)=\frac{y_1^2}{4}+\frac{y_2^2}{9}-1$. $S=\{(y_1,y_2) |F(y_1,y_2)=0\}$, and $D=\{(y_1,y_2) |F(y_1,y_2)<0\}$. I want to show ...
22
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1answer
310 views

Does a set $A \subseteq [0,1]$ exist such that $A$ is homeomorphic to $[0,1] \setminus A$?

Does a set $A \subseteq [0,1]$ exist such that $A$ is homeomorphic to $[0,1] \setminus A$? I have no idea how to attack this problem. Any help will be appreciated.
2
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0answers
40 views

quotient by contractible space and homotopy equivalent? [duplicate]

Let $X$ be a space and $L$ be a subspace of $X$ that is contractible. Then is $X/L$ homotopy-equivalent to $X$ itself? it doesn't seem trivial to me at al...If so, how to show it rigorously? Could ...
0
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1answer
61 views

Subsets of the set [0,1]x[0,1] with the dictionary order topology

I'd appreciate some feedback on whether or not I'm thinking about this problem correctly. Show whether or not the subsets $Y \subset [0,1]$x$[0,1]$ = X are open in X when X has the dictionary ...
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0answers
53 views

Let the intervals [0, 1] and (0, 1) be given the standard ordering…

Let the intervals [0, 1] and (0, 1) be given the standard ordering. Determine whether each of the following subsets Y ⊂ X is open in X when X has the dictionary order topology. In each case, either ...
2
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1answer
24 views

Boundary bumping theorem

Boundary bumping theorem: Let $X$ be a compact connected space, $A$ its closed proper subspace, $C$ a component of $A$, then $C ∩ ∂A ≠ ∅$. ($∂$ means boundary.) I wonder if the compacness assumption ...