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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2
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1answer
32 views

Extending a homeomorphism of the open disk to the boundary.

Let $D^2 = \{x \in \mathbb{R}^2 : ||x||\leq 1\}$ denote the closed disk and $int(D^2)$ denote its interior. If I have a homeomorphism $\ f: int(D^2) \rightarrow int(D^2)$ it is clear that it is not ...
0
votes
1answer
40 views

Is the function $f:\mathbb{R} \longrightarrow \mathbb{R}^\omega$ defined by $f(t) = (t, t, t, \ldots)$ continuous in the uniform topology?

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Notation used: Countably ...
4
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0answers
52 views

In which of the three topologies is $f(t)=(t, 2t, 3t, 4t, \ldots)$ continuous? Here, $f$ is a function from $\mathbb{R}$ to $\mathbb{R}^\omega$.

Can someone please verify my proof or offer suggestions for improvement? Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of the three topologies is $f(t)=(t, 2t, ...
0
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1answer
47 views

Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
3
votes
1answer
47 views

Closed surjection that does not preserve regularity

Def Map $p\colon X\rightarrow Y$ is perfect if it is a closed surjection and $p^{-1}\left(\left\{y\right\}\right)$ is compact for each $y\in Y$ It is well known that perfect maps preserve regularity, ...
1
vote
1answer
19 views

Under what terms does $r$-closedness depend on the base

Let $B_n$ form a subbase of a topological space $X$, and let $A\subset X$. One way to define the closedness of $A$ is to say that if $x$ satisfies that any arbitrary intersection of sets from $B_n$ ...
1
vote
1answer
41 views

Choice of Metric Gives Nice Topological Properties

I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices. Nice is defined as compact, Hausdorff, or whatever ...
2
votes
1answer
42 views

Proving that a quotient space is compact but not Hausdorff

Let ∼ be the equivalence relation on $\mathbb{R^2}$ defined by $(x, y) ∼ (x_0 , y_0 )$ if and only if there is a nonzero $t$ with $(x, y) = (tx_0 , ty_0$ ). Prove that the quotient space ...
0
votes
1answer
35 views

Prove that real line and parabola are topologically same but different geometrically?

As far I know I need to find a homeomorphism. So if I consider a real line then $f: \Bbb R \to P$ defined by $f(x)=(x,x²)$ where $P=$ parabola. I have problem with continuity of $f$ and $f^{-1}$.
2
votes
1answer
76 views

What is a Topological Group Intuitively?

What is a topological group intuitively, beyond just being able to say things are near to each other in a group, and why is it a good idea to consider this theory as part of general topology as ...
5
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1answer
38 views

Smooth embeddings of the $2$-sphere

I have a past qual question here: given a smooth embedding $f \colon S^2 \to \mathbb{R}^3$, show that there must exist distinct points $p,q \in S^2$ such that the tangent planes to the embedded sphere ...
2
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0answers
26 views

Comparing product topologies

Let $C$ denote the set of complex numbers. Let $T$ be the smallest topology on $C$ such that singletons are closed. Let $T_1$ denote the smallest topology on $C$x$C$ such that all the polynomials in ...
0
votes
2answers
52 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
1
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2answers
40 views

Connected Sets in Topology

Theorem: Let $(X,\mathscr{T})$ be a topological space. If $E$ is connected and $K$ is such that $E\subseteq K\subseteq\mathrm{cl}(E)$, then $K$ is connected. (Cl(E) is closure of E) Question: ...
0
votes
1answer
70 views

Is the sphere $S^n$ always arcwise connected?

I have a small question about the connectedness of the sphere; Is the sphere $S^n$ always arcwise connected ? Thank you.
0
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0answers
33 views

On Compact Open Topology

Consider $X$ as a compact topological space and $Y$ as a metric space. Consider $C(X,Y)$, the set of all continuous functions from $X$ to $Y$. Prove that $C(X,Y)$ with compact open topology is induced ...
2
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1answer
41 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
2
votes
1answer
42 views

Topology of a nested sequence of subsets

Hi everyone I'd like to know if the following proof is correct, I think so. And also if there is a more direct approach without the many subcases. Thanks in advance Let $X$ be an infinite set, and ...
2
votes
0answers
26 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
3
votes
2answers
76 views

Is every dense subspace of a separable space separable?

If $X$ is a separable topological space, and $V$ is some dense subspace of it, is $V$ necessarily separable?
3
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0answers
37 views

Fundamental group of 7-gon with labelling scheme $abaaab^{-1}a^{-1}$

I want to calculate the fundamental group of of a $7$-gon with labelling scheme $abaaab^{-1}a^{-1}$. I will call the quotient space $X$ with reference point $x_0$. This is what I tried: If you have ...
0
votes
0answers
29 views

Proof: Product Topology Question XxY

If $f$ is maps from topogical spacce $Z$ to $X\times Y$ so: $f$ is continuous iff : $\begin{cases} (p_X)\circ f: Z \rightarrow X\times Y \rightarrow X \\ (p_Y)\circ f: Z \rightarrow X\times Y ...
1
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2answers
30 views

Set of real symmetric matrices with signature $(2,1)$ is open

Let $S$ be the space of all $3\times 3$ real symmetric matrices, let $B$ be the subset of $S$ with signature $(2,1,0)$. Show that $B$ is open in $S$ in the topology of $\mathbb{R}^6$. My thoughts: ...
1
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4answers
49 views

Prove that: If $X$ is a topological space and $A$ and $B$ are two subsets of $X$ then,$Cl(A) \cup Cl(B) = Cl(A \cup B) $

Prove that: If $X$ is a topological space and $A$ and $B$ are two subsets of $X$ then,$Cl(A) \cup Cl(B) = Cl(A \cup B) $ where $Cl(H)$ means the closure f the subsets $H $of $X$. I was able to prove ...
2
votes
1answer
43 views

Boundary connected sum of manifolds

I have two related questions about the boundary connected sum of manifolds with boundaries. Let $T=S^1 \times S^1$ be a torus and let $X=T \times [0, 1]$ be the cylinder over the torus. Let $X'$ be a ...
1
vote
3answers
102 views

Exponetial map from real line to circle

Is the map $x\to e^{ix}$ from real line $\Bbb R$ to circle open? If I take any closed or half closed subset instead of $\Bbb R$ then this is definitely not open. But I'm little bit confused when ...
0
votes
0answers
52 views

A Property of Baire Spaces

Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the ...
0
votes
1answer
38 views

ω1 disconnected

What is the reason that $ω1 (ω1 + 1)$ is disconnected? My idea is that we know every uncountable well order set contains a copy of ω + 1 as an initial segment. So for ω1 (ω1 + 1), we can find the ...
3
votes
1answer
84 views

Non Archimedean field

Prove that there exist an ordered field which is not complete but in which every cauchy sequence has a limit? Here Completeness means every bounded above subset has a least upper bound. And for cauchy ...
1
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1answer
45 views

Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
0
votes
2answers
38 views

Liftings in Topology

I am wondering about this: Assume you have a class $[f] \in \pi_1(B,b_0)$ and a covering map $p:E \rightarrow B$. Now, I know that if you take any two paths $g,h \in f$ that are homotopic and they ...
0
votes
3answers
28 views

Basic question about limit points.

Given a set $E=(0,1)$, what is the set of limit points for this set? I read somewhere that the set of limit points consist of only 2 elements: 0 and 1. I can understand that these are two limit points ...
3
votes
1answer
94 views

Set of limit point is a closed set. [duplicate]

Let $L$ be the set of limit points of set $A$. How do you show that the set $L$ is closed? Or, is this statement not necessarily always true?
0
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0answers
42 views

Is Hausdorff necessary condition for Arzela-Ascoli? [duplicate]

Here is a theorem proven in Munkres-Topology Let $(X,\tau)$ be a nonempty compact space. Let $d$ be any metric induced by a given norm on $\mathbb{R}^n$. Let $\overline{\rho}$ be the ...
0
votes
3answers
49 views

Closure of a subset of a metric space is closed

From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$. I need to ...
0
votes
2answers
27 views

Topological isomorphism vs isometric isomorphism

We say that: $T:(X,\|\cdot\|_X)\rightarrow (Y,\|\cdot\|_Y)$ is a isometric isomorphism if it is a linear isomorphism, and it is an isometry, that is $\|T(x)\|_Y=\|x\|_X\quad \forall x\in X;$ ...
1
vote
0answers
26 views

Canonical choice of inverse system for profinite set.

Let $X$ be a profinite set - an inverse limit $\varprojlim X_i$. How can one prove that then $X=\varprojlim Y_i$, where $Y_i$ is finite quotient spaces of $X$? I may prove it if $X$ is topological ...
4
votes
3answers
81 views

Is a direct proof of this possible

Consider the following statement $x_n \to x$ if and only if every subsequence of $x_n$ has a subsequence that converges to $x$. $\implies$ is clear. A proof of the other direction is given here. ...
2
votes
1answer
34 views

Simple question on connectedness in a subspace [duplicate]

For some reason I am having some trouble on this basic point set topology question: Suppose $X$ is connected, and $A$ is a connected subset of $X$, and that $B$ is a clopen set in $X-A$ (not in $X$, ...
1
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1answer
38 views

Munkres topology page 153

In Munkres p.153, we have a proof like this He mentions that $B_0$ is open, so there is some interval $(d, c]$ containing $c$, which is contained in $B_0$. So we know if $c=b$, $(d, c]=(d, ...
0
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0answers
43 views

Completation of an n.v.s. and dimensions of subspaces.

I don't know if the following statement is true: Let $X$ be an n.v.s. with $\text{dim}(X)=\infty$ and not Banach; and $\bar X $ its completation in the bidual space. Let $Y$ be a closed subspace ...
0
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1answer
49 views

To show closed subset of $R^2/{\sim}$ contains origin

I am working on the old preliminary exam from my university. I found trouble to solve the following problem. Could you please help me on it. Let $X = \mathbb R^2/{\sim}$ be quotient space where $x\sim ...
1
vote
1answer
42 views

Separability of the space of bounded uniformly continuous functions

Let $(X,\rho)$ a metric space. Do the space $U_b(X)$ of uniformly continuous and bounded real-valued functions on $X$ is separable? It seems that the point is to pass through the Stone-Cech ...
0
votes
0answers
28 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
1
vote
1answer
59 views

Gluing the ends of a cylinder. Can we get other than a torus?

Let $X=S^1 \times I$ be a cylinder, where $S^1$ is the 1-dimensional circle. If we glue the "bottom" boundary $S^1 \times 0$ and the "top" boundary $S^1\times 1$ by a homeomorphism sending $x\times 0$ ...
1
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1answer
34 views

Induced topology vs subspace topology

Reading my book I found this definition of an induced topology, which was then alleged to be equivalent to the standard definition of the subspace topology for that special case. However, I'm failing ...
1
vote
1answer
29 views

A question on metrizability

In Munkres, Topology, there is a theorem 10.3 Let $f:X\to Y$ Let X be a matrizable. The function f is continuous iff for every convergenct sequence $x_{n}\to x$ in X, the sequence $f(x_{n})\to f(x)$. ...
3
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1answer
52 views

Disconnected Topoological Space with Intermediate Value Property

Does There exist a disconnected topological space with intermediate value property? Intermediate Value Property states that 'a topological space X is said to have intermediate value property if for ...
1
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3answers
152 views

Let $X$ be a metric space with metric $d$. Show that $d:X \times X \longrightarrow \mathbb{R}$ is continuous.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I want help with my proof in particular. Let $X$ be a metric ...
0
votes
2answers
24 views

Nullhomotopic map extended

I have troubles understanding this proof: Let $h:S^1 \rightarrow X$ be a continuous map, then we have that if $h$ is nullhomotopic, $h$ can be extended to a continuous map $k:B^2 \rightarrow X.$ ...