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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1answer
26 views

The interval $(0,\infty)$ is an open set.

I want to prove this using interior points, $\epsilon$-neighborhoods and interior sets. The interior of a set A is denoted $A^o$. To show that $(0,\infty)$ is an open set, we must show that ...
1
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1answer
39 views

Ambiguity definitions - accumulationpoint

The literature is a bit ambiguous in my point of view. Limit points and accumulation points seems to be the same. I can accept that; that's just two names for the same. But I've seen different ...
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1answer
26 views

Alexander–Briggs notations for the links or knots of $N^3_m$

We can use Alexander–Briggs notations for the links or knots. For example, is three separate loops with no links. And there are many other examples of Alexander–Briggs notations for three ...
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1answer
21 views

Show that span is separable [on hold]

Let $X$ be a n.v.s and $A\subset X$. Show that if $A$ is enumerable then $\overline{ \text{span}\{A\}}$ is separable
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1answer
26 views

Is a mapping f bijection ?

If Y is an one point compactification of X,Y=X union {p}, p not belong to X. Is a mapping f from X into Y bijection? If it is not, what are the assumptions I add to be f bijection ? Thanks for any ...
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1answer
45 views

Is it true there exists $f:S^{2n}\longrightarrow S^{2n}$ making the diagram commutative?

Let $g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$ be a continuous map where $\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists $f:\mathbb ...
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1answer
48 views

Covering map of $\mathbb R \mathbb P^2$

The question I am trying to answer is: Does the quotient map $ q:[0,1] \times [0,1] \to \mathbb R \mathbb P^2$ extend to a covering map $\mathbb R^2 \to \mathbb R \mathbb P^2$ I know that the ...
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1answer
36 views

Existence of particular open subgroups, given a prof-finite group

I have currently read a proof (existence of sections for pro-finite groups (in the book profinite groups of Ribes)) and I did not understand the following two facts used (without mentioning any ...
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1answer
42 views

Link between a topological space and a manifold

A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that, (i) any union of these subsets is one of the subsets. (ii) any ...
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1answer
34 views

weak-* topologies

Say $S = \{z \in \ell_\infty : z_n \in \{0,1\}\}$. Suppose I am asked a question about the weak-* topology on $S$. How am I supposed to make sense of this? The weak-* topology is a topology on a dual ...
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0answers
23 views

Can we say that $[0,\omega_1]$ is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

Let $E$ be a topological space, $q \in E$. The neighbourhood point game $G_{np}(q,E)$, is defined as follows. It is played by two players, ONE and TWO.In the n's step $n \in \omega$, ONE chooses ...
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0answers
23 views

Finding a uniformly continuous function from a non complete space to $\mathbb{R}^{+}$

Assume that $(X, d)$ is not complete. Prove that there exists a uniformly continuous function $f:X \rightarrow \mathbb{R}^{+}$ such that $\inf_{X} f(x) = 0$. We are guaranteed the existence of a ...
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2answers
18 views

Cantor Intersection Theorem in $R^n$

I am looking at the Cantor Intersection Theorem from Apostol's Mathematical analysis. Let {$Q_1, Q_2, $...} be a countable collection of nonempty sets in $R^n$ such that: 1) $Q_{k+1} \subset Q_k$ ...
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0answers
45 views

Showing $\mathbb{R}$ is a completion of $(\mathbb{Q}, | \cdot |)$

Recall the definition of a completion: A complete metric space $(Y, d)$ is said to be a completion of another metric space $(X, d)$ if there exists a map $f: X \rightarrow Y$ such that $f$ is an ...
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1answer
39 views

difference between sequence in topological space and metric space

Reading the book about topology, I find an interesting difference between two spaces: We use net convergence $\{p_\lambda\}_{\lambda\in\Lambda}\rightarrow p$ in frontier, but use sequence ...
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1answer
20 views

Inner product space is connected

How does one show any inner product space is connected? Shall I start with assuming that it is not connected, and arrive at a contradiction? So let $X$ be an inner product space, and there exist open, ...
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1answer
33 views

“internal” definition of complete regularity?

There is something strange (I think) about the complete regularity separation axiom. Consider the definitions. T0 means for every two distinct points there is an open set containing exactly one of ...
4
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2answers
40 views

A zero-dimensional Hausdorff space is totally disconnected

The full question: A space is zero-dimensional if the clopen subsets form a basis for the topology. Show that a zero-dimensional Hausdorff space is totally disconnected. Recall a space is totally ...
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1answer
33 views

Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
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0answers
40 views

A lemma on function spaces

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me? Lemma: let $X$ be in |SET| $(Y, d)$ in |MET|, $f_n$, $f$ is in $Y^X$. Then $f_n\to ...
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0answers
22 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
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2answers
53 views

How do I prove this set is connected?

Define $A=\{(x,y):y=\sin(1/x), x\neq 0\}$ and $B=\{(0,y):-1\leq y \leq 1\}$. How do I prove that $A\cup B$ is connected? I can see this is not path connected but cannot prove why it is connected..
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1answer
51 views

Open and Closed Covering [on hold]

Suppose $p:\widetilde{X} \mapsto X$ is a covering with $f,g:Y \mapsto \widetilde{X}$ continuous such that $pf = pg$. Why is the set of points in $Y$ for which $f=g$ open and closed in $Y$?
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1answer
68 views

Dense sequence in $[0,1]$

There is the theorem proved by Liouville which states that if $x$ is irrational then there are infinitely many fractions $\frac{p}{q}$ such that $|x-\frac{p}{q}|<\frac{1}{q^2}$, i.e. ...
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2answers
55 views

Limit of a sequence and a closed set

It's a dumb question, but I need to assure myself: If $V$ is a subset of a metric space $W$, then if we take a sequence in $V$ and it has a limit in $W\setminus V$, does it mean that $V$ is not ...
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1answer
28 views

Uniform convergence of sequence of functions on compact set [on hold]

Let $X$ be a compact topologial space, $U$ an open subset of $\mathbb R$ and $f:X\to\mathbb R$ a continuous function such that $f(X)\subseteq U$. Prove that if a sequence of functions $f_n:X\to\mathbb ...
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0answers
42 views

Connected space whose every subspace is disconnected

We know that a subspace of a connected space can be disconnected eg. $\mathbf{Q} \in \mathbf{R}$ where $\mathbf{R}$ is connected but $\mathbf{Q}$ is totally disconnected as a subspace. My question ...
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0answers
61 views

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ what does this mean?

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ I saw this in my topology assignment. The question was about quotient spaces and homeomorphisms. I have never seen this expression before so it doesn't ...
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0answers
10 views

Criteria to judge te quality of a journal [migrated]

Is a journal with high impact factor is better than that with low impact factor? Is impact factor is right criteria to judge the quality of a journal?
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1answer
39 views

Topological Structure of Finite Set

I encounter with a problem in Topological Manifold written by Lee: How many different topological structure of $\{1,2,3\}$? It is easy to make a list of the question, and the answer is $9$. ...
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0answers
26 views

$L \cap V = \overline{L} \cap V \implies L$ open in $\overline{L}$. [Solved]

how do I prove this? My attempt: Let $x \in L$, and for some open set $V \ni x$ suppose that $L \cap V = \overline{L}\cap V$, also maybe use the fact that $L \cap V$ is closed in $V$. If $x \in L$ ...
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1answer
15 views

$L \cap V = \overline{L} \cap V$ when $L\cap V$ is closed in $V$.

Let $E$ be a topological space and $L, V \subset E$, $V$ open, and $L \cap V$ closed in $V$, then $\overline{L} \cap V = L \cap V$. Attempt: $L \cap V$ closed in $V$ implies $L \cap V = F \cap V$ for ...
2
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1answer
28 views

Path components of Wedge Sum

I couldn't find this anywhere else, so I decided to post it here. I suspect that the wedge sum $⋁X_α$ of pointed spaces $X_α$ has as path components all components of the topological sum $\oplus X_α$ ...
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0answers
31 views

Nagata Smirnov Metrization Theorem

I am looking for a proof for Nagata-Smirnov Metrization Theorem, but I couldn't find one that is readable. I found the paper by Nagata written in 1954 but it is unreadable and uses old notation. ...
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1answer
28 views

If a subset $E$ of $R^n$ is bounded then E is totally bounded

I am trying to prove the above proposition. The book that I am looking at contains E in a cube of the form T=[−b,b]×⋯×[−b,b] for some large b>0. Then, since any subspace of a totally bounded metric ...
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0answers
21 views

Linear order set

I´m trying to prove the following statement: if (T,<) is a linear dense order and it is CCC (i.e. every disjoint family of non-empty open sets is countable), then it is Lindelof. Thankyou for your ...
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3answers
38 views

Non- compact set

We have to show that $$S = \lbrace (x,y) \in \mathbb{R}^2 : xy>1 , x^2 + y^2 < 5 \rbrace $$ is non-compact . I tried taking an open cover $$x^2 + y^2 = 6$$ and then took countable subcover ...
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0answers
38 views

A subset E of $R^n$ is totally bounded if and only if E is bounded

I am studying Compactness in metric space with Gamelin and Greene's Introduction to Topology and am confused about lemma 5.4 in the book. A metric space $X$ is totally bounded if for each $e > 0$, ...
3
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1answer
86 views

Is there an injective continuous map $\mathbb{R}^2 \rightarrow \mathbb{R}$?

It is commonly known fact that there exists a continuous surjective map $\mathbb{R} \rightarrow \mathbb{R}^2$. So it bids to ask: Is there an injective continuous map $\mathbb{R}^2 \rightarrow ...
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6answers
52 views

To show a set is open

Given $A \in \mathbb{R}$ be open define $B = \lbrace{(x,y) \in \mathbb{R}^2 : x \in A} \rbrace$ Show that $B$ is open in $\mathbb{R}^2$
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1answer
40 views

How to show that every Suslin tree is Frechet-Urysohn

A topological space $X$, is Frechet-Urysohn, if, given $x \in \overline A \subset X$, there exists a sequence of points in $A$ which converges to $x$. I am trying to prove that every Suslin tree, is ...
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0answers
10 views

is there any relation between category theory and differentiation theory in locally convex spaces

Is there any relation between category theory and differentiation theory in locally convex spaces. If it so , what type of relation.can you conclude in one or two lines
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32 views

Sequence of increasing compact sets

Suppose $X$ is a locally compact metric space which is $\sigma$-compact. Let $K$ be a compact subset of $X$. We can find a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + ...
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1answer
25 views

Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...
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3answers
119 views

Cantor's Teepee is Totally Disconnected

Let $C^\prime$ be the Cantor set and let $C = C^\prime \times \{0\}$ (viewed as a subset of $\mathbb{R}^2$). For $c \in C$, let $L(c)$ denote the half-closed line segment connecting $(c,0)$ to ...
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2answers
43 views

Give an example of a non-separable subspace of a separable space

I'm trying to find an example of a non-separable subspace of a separable space. What kind of examples are there?
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1answer
34 views

Existence of a neighbourhood of a compact set ( from james fibrewise topology)

I'm reading James' Fibrewise topology book and I'm trying to understand the proof of proposition 7.4 , it says: Let X be a proper G-space . Then X is fibrewise regular over X/G. Proof For any $x \in ...
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2answers
64 views

The set $S=\{(x,y) \in \mathbb{R}^{n} \times \mathbb{R}^n = \mathbb{R}^{2n} ; x \neq y\}$ is connected if $n \geq 2$.

When n = 1 it is easy to see that is not connected, it just take the split open $ S=U_1 \cup U_2$ such that $U_1 = \{(x,y) \in \mathbb{R}^2 ; x > y\}$ is $U_2 = \{(x,y) \in \mathbb{R}^2 ; x < ...
3
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0answers
47 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
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1answer
50 views

Show that $f(\operatorname{int} A) = \operatorname{int}(f(A))$ when f is a homeomorphism.

Show that $f(\operatorname{int} A) = \operatorname{int}(f(A))$ when f is a homeomorphism. My trial is as follows. It seems to me that there is an error or it needs more detail. Any answer or ...