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Questions tagged [general-topology]

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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13 views

Topology From Sequences

Let $X$ be a set and $f:X\to\mathcal{P}(X^\omega)$. Under which circumstances is there some topology $\tau$ on $X$ such that $f$ maps each point $x$ to the set of converging sequences in $\tau$ with ...
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1answer
13 views

If $D$ is a ultrafilter on $I$ and $(a_i) \mapsto_{D} a$ and $f_i \mapsto_{D} f$ then $ Sup_{x}f_i(x, a_i) \mapsto_{D} Sup_{x} f(x, a)$

Let $X$ be a topological space and let $(x_i)_i \in I$ be a family of elements of $X$. If $D$ is an ultrafilter on $I$ and $x \in X$, we write $$x_i \mapsto_{D} x$$ if for every neighborhood $U$ of $x$...
1
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1answer
22 views

Cartesian Product of Singleton and $\mathbb{R}$ Open In $\mathbb{R}^2$?

I'm having difficulty seeing how $X=\{x\times y:y=0\}$ is open in $\mathbb{R}^2$. Being open would imply for any point $x\in X$, I can find a neighborhood around it contained in $X$, but I don't think ...
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0answers
11 views

$\lambda$-Lemma or Inclination Lemma

I appreciate it if someone provide some intuitions for the $\lambda$-Lemma (Inclination Lemma) in dynamical systems? I am trying to think about a pictorial example, but I am having a hard time.
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2answers
15 views

Is a manifold $N$ smoothly embedded in a manifold $M$ of the same dimension open in $M$?

Consider a manifold smooth manifold $N$ smoothly embedded in another manifold $M$ of the same dimension. Is it true that $N$ is open in $M$? I think this is true, due to the open mapping theorem. If ...
1
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1answer
23 views

Find the equivalence class of $X\coprod Y$

Define $i_X:X\rightarrow X\coprod Y$ as the inclusion mapping $i_X(x)=x$, and similarily $i_Y:Y\rightarrow X\coprod Y$ as the inclusion mapping $i_Y(y)=y$. Let $X$ and $Y$ be topological spaces....
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0answers
9 views

Dimension of Unstable Manifolds of the Equilibrium Points Connected by a Heteroclinic Orbit

Consider the ODE \begin{align} \dot x = f(x) \tag{1} \end{align} Let $x_0$ and $y_0$ be hyperbolic critical elements (fixed points or periodic solutions) of $(1)$, and let $W^u$ and $W^s$ denote the ...
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5answers
13 views

Proving that $L = \{(x,y)\in\mathbf{R}^2:y=mx+c\}$ is closed in $\mathbf{R}^2$.

I am required to prove that the set $L = \{(x,y)\in\mathbf{R}^2:y=mx+c\}$ where $m,c\in\mathbf{R}$ is closed in $\mathbf{R}^2$. I have tried to show that $\mathbf{R}\backslash L = \bigcup_{(a,b)\in\...
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0answers
13 views

$\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\Omega)$

We have the following that is derived from the Bramble-Hilbert Lemma $\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\...
3
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0answers
29 views

Locally convex topological space

I got a problem with this task. Let $M=\{[f] , f:[0,1] \to \mathbb{F} , \int_0 ^1 |f(x)|^x dx<\infty \}$, where $f$ is a measurable function, be a set and let $\rho(f,g)=\int_0 ^1 |f(x)-g(x)|^x dx$ ...
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0answers
18 views

Lipschitz continuity of a function on convex sets

I have an open bounded convex set in $\mathbb{R}^n,$ $A.$ Fix $x\in A,$ and consider the function $\phi:S^{n-1}\to\mathbb{R}$ such that, for all $e\in S^{n-1},$ $\phi(e)=\sup\{t\geq 0: x+te\in A\}.$ I ...
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0answers
11 views

How to prove the relationship between krasnoselskii's genus and the dimenson of a vector space?

I have to work with this definition for Genus: Let us denote by $U$ the class of all closed subsets $A ⊂ X- \{0\}$ that are symmetric with respect to the origin, that is, $u ∈ A$ implies $−u ∈ A$. ...
0
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1answer
24 views

Open balls appearance

We know that with the euclidean metric the open balls in $\mathbb{R}^2$ are circles without the frontier, of course. My question is if there exist a known metric in $\mathbb{R}^2$ such that the open ...
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1answer
18 views

The non-empty intersection of two open discs contains an open disc.

Is the following argument correct? Let $D_1$ and $D_2$ be any open discs in $\mathbf{R}^2$ with $D_1\cap D_2\neq\varnothing$. If $(a,b)$ is any point in $D_1\cap D_2$, show that ther exists an open ...
0
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1answer
20 views

Ultrafilters over product spaces

Suppose that for $i\in I$, $X_i$ are topological spaces and $U_i$ is an ultrafilter over $X_i$. Consider the space $\Pi_i X_i$ with the product topology. I want to know when, if ever, it is possible ...
3
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2answers
33 views

Compact refinement of a covering

Suppose $X$ is compact. $A$,$B$ open sets which cover $X$. Can $X$ be covered by compact sets $C$,$D$ such that $C \subseteq A$ and $D \subseteq B$?
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0answers
32 views

Open sets in infinite dimensional spaces

Let $C$ be an closed subset of the Banach space $X$. I am wondering whether the following statements are equivalent for any Banach space: $O$ is an open set containing $C$. For every $x\in \partial C$...
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1answer
22 views

Proving that a unit disc is open in $\mathbf{R}^2$.

I am trying to prove part $(i)$ here. The following is my attempt at the problem is it correct? Proof. Let $\alpha = (a,b)\in D$ and $\beta = (x,y)\in R_{(a,b)}$ and $O = (0,0)$, then given the ...
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1answer
17 views

Convergence of a sequence for the three-point set

In Munkres Topology section 16 in the subsection on Hausdorff Spaces there is a motivating example involving the three-point set $\{a, b, c\}$ which states that the sequence defined by setting $...
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0answers
13 views

Draw a map on the projective plane

Draw a map on the projective plane with five regions, all of which are adjacent to each other. (This proves that the chromatic number of the projective plane is ≥ 5.)
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2answers
38 views

Defining a set which contains one element per element in another set

I'm doing a bunch of topology exercises at the moment, and one thing I sometimes want to express is "given set $X$, let $Y$ be a set which contains an element per element in $X$". As an example, here'...
0
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2answers
29 views

Sequence converges iff function is continuous on extension of natural numbers (topology)

Let $(X,d)$ be a metric space, and $\{x_n\}$ be a sequence given by $x:\mathbb{N}\rightarrow X$. Show $\{x_n\}$ converges to $c\in X$ if and only if $$f:\mathbb{N}\cup\{+\infty\}\rightarrow X \ ...
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2answers
22 views

Intersection of sequence of compact non empty sets has a similar diameter to the elements in the sequence?

Suppose $(K_n)$ is a sequence of nested, compact, nonempty sets, where $K_1 \supset K_2 \supset K_3 \supset \cdots,$ and $K = \bigcap_{n \in \mathbb{N}} K_n.$ If for some $x > 0,$ $\operatorname{...
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3answers
27 views

How does this inequality hold? $|d(x,x_0)-d(y,x_0)|\le d(x,y)$

I came across this during some reading: $|d(x,x_0)-d(y,x_0)|\le d(x,y)$. I can't seem to figure out why it holds. Here $d$ is a metric.
5
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1answer
264 views

Arc length of nowhere differentiable arcs.

Does there exist an arc (something homeomorphic to the interval $[0, 1]$) that is nowhere differentiable? If so, how does one define the arc length function along such an arc? Can a topological arc ...
1
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1answer
25 views

Convex sets and semicontinuous functions

I have this problem: Let $X$ be an open bounded subset of $\mathbb{R}^n,$ and fix $x_0\in X.$ For all $e\in S^{n-1}$ we put $$\phi(e)=\sup\{t\geq0: x_0+te\in X\}$$ $$\overline{\phi}(e)=\inf\{t\geq0: ...
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2answers
51 views

Prove that: $\bigcap_{t\in T} \space I_t \neq \varnothing $ for non-disjoint family of intervals from $\mathbb{R}$.

Let $ (I_t)_{t \in T} $ be an indexed family of intervals of $\mathbb{R}$ that are closed, bounded and mutually non-disjoints. Prove that: $$\bigcap_{t\in T} \space I_t \neq \varnothing $$ ...
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1answer
14 views

Question about proof of Heine-Cantor (i.e. compact and continuous implies uniform continuous)

If anyone has seen the wikipedia page for the Heine-Cantor theorem, I find something off about the proof it presents. It would be incredibly tedious to write it all out here because it's pretty ...
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3answers
52 views

For nonempty sets $A \subset M, B \subset N$, if $A \times B$ is compact, then $A$ and $B$ are compact.

Consider any pair of sequences $(a_n) \subset A$ and $(b_n) \subset B.$ The sequence $((a_n),(b_n)) \subset A \times B$ is contained in $A \times B.$ Hence, there is a subsequence $(a_{f(n)}, b_{g(n)})...
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0answers
19 views

Proper map induced from a continuous map

Consider the following claim. Claim: Let $X$ and $Y$ be locally compact topological spaces, and $f : X \longrightarrow Y$ a continuous map. Suppose that $W \subset X$ is a relatively compact set. ...
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2answers
46 views

Munkres topology question §25 number 5: Show that a space is not locally connected

Let $X$ be the rational points of the interval $[0,1] \times 0$ in $\mathbb{R}^2$ and define the point $p = 0 \times 1$. Let $T$ be the union of all line segments joining $p$ with a point of $X$. Show ...
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1answer
51 views

Re: Stewart & Tall, “Complex Analysis” ($2^\text{nd}$ ed. 2018), Proposition 2.45: constructing a family of space-filling curves

Here are the statement and proof of the proposition, as they appear in the book: Let $\mathbb{U}^2$ be the unit square $\{ x + iy : 0 \le x \le 1, \ 0 \le y \le 1\}$. [$\ldots$] PROPOSITION 2....
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4answers
238 views

Is the torus with one hole homeomorphic to the torus with two holes?

I would like to understand why the torus with one hole is not homeomorphic to the torus with two holes. I have a very basic understanding of the concepts (I know what an homeomorphism is but not much ...
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0answers
29 views

Family of embeddings with compact support is isotopic?

I am currently reviewing materials in smooth manifold and stuck at this problem. Let $f_t : N \rightarrow M$ be a family of embeddings: $f_t$ is an injective proper immersion for each $t \in [0,1]$. ...
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0answers
22 views

Let Y be an ordered set in the order topology. Let $f,g:X→Y$ be continuous. Show that the set $[{x|f(x)≤g(x)}]$ is closed in X.

Let Y be an ordered set in the order topology. Let $f,g:X→Y$ be continuous. Show that the set $[{x|f(x)≤g(x)}]$ is closed in X. My Try : $[{x|f(x)>g(x)}]=∪_{y∈Y}[{x|f(x)>y>g(x)}]$ and $[{x|...
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1answer
24 views

Proof verification on the openness of the space of matrices of full rank

Let be $m<n$ and $M_m(m\times n, \mathbb{R})$ be the set of the matrices $m \times n$ of full rank $m$. I want to show that $M_m(m\times n,\mathbb{R})$ is an open subset of $M(m\times n,\mathbb{R}...
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1answer
42 views

Points of continuity in $ M_n (k) $ of minimal polynomials

The goal is to show that $\Gamma $ the set of points of continuity of $ M \mapsto \pi_M $ is $ \{ M \in M_n(k) , \chi_M = \pi_M \} $ . Where $ \pi_M $ is the minimal polynomial of $M$ and $ \chi_M $ ...
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1answer
19 views

Elements in the basis of Product topology determined by sub-basis other than sub-basis elements.

I could prove the result for $|\Lambda|$ finite. Here $|\Lambda|$ is arbitrary. My attempt:- Let $\langle x_{\alpha}\rangle_{\alpha\in \Lambda}\in B \implies \langle x_{\alpha}\rangle_{\alpha\in \...
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3answers
82 views

Connectedness of $[0,1) \times [0,1]$.

I'm taking an introductory course in topology and I came across the following problem. Decide if $X=[0,1) \times [0,1]$ with the topology from the lexicographical order is connected or not. When ...
3
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1answer
31 views

A space is $T_0$ if and only if it is homeomorphic to a subpspace of $S^I$ with $S$ the Sierpinski space for some $I$ [Proof Verification]

As the title states, I want to check my proof of the following: Proposition. A topological space $(X, \tau)$ is $T_0$ if and only if there exists a set $I$ such that $(X,\tau)$ is homeomorphic to a ...
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1answer
56 views

(Baby Rudin) Thm 2.34: Compact subsets of metric spaces are closed.

At the beginning of proof, $V_q$ should be $V_p$, so it seems to be just typo. Am I right? And I have understood up to the part $V=V_{q1} \cap \cdots \cap V_{qn}$ but cannot understand why the $V$ is ...
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0answers
30 views

Problem 1.3.2 in Guillemin and Pollack.

The question and its answer is given below: But I could not understand 1- why he defined $\psi(\tilde{U}) = U$? I know that the importance of $\tilde{U}$ to adjust the dimension. 2-why $\psi ^...
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0answers
28 views

“Schaums Outline of General Topology” by S. Lipschutz. Is it good choice for self study General Topology?

I am looking for a book to self study general topology. I found "Schaums Outline of General Topology" by S. Lipschutz. Is it good choice ? Many people recommend other books (for example Munkres "...
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1answer
36 views

Reference for the “points of continuity of a function is a $G_\delta$ set”.

Let $f: X \to Y$ be a function between metric spaces. I was told that the points of continuity of $f$ are a $G_\delta$ set and the points of discontinuity an $F_\sigma$ set of $X$. Can anyone give me ...
0
votes
1answer
49 views

a sequence of compact converging to another compact

Let $K$ be a compact and $A_{n}$ the set of points having a distance equal to $\dfrac{1}{n}$ to $K$. Suppose the interior of $K$ is empty. Can we say that for each $x$ in $K$ there is a sequence $(y_{...
0
votes
1answer
11 views

Construct Compact Exhaustion using Paracompactness

Let $M$ be a topolgical $n$-manifold. I have to show that there exist a sequence $(K_i)_{i \in \mathbb{N}}$ of compact subspaces $K_i \subset M$ with properties $K_i \subset K_{i+1} $ for all $i \in \...
2
votes
0answers
40 views

Invariance of boundary $S^1\to S^1$

Show that every homeomorphism $f: D^2\to D^2$ restricts to a homeomorphism $f_{|\partial D^2}: \partial D^2\to\partial D^2$ I want to proof the following statement. I have to show, that $f_{|\partial ...
2
votes
3answers
35 views

Proving that a non-empty countably infinite subset of $\mathbf{R}$ is not open.

I am require to prove the following propostion. Proposition. Given a non-empty countably infinite subset of $\mathbf{R}$ say $F$, it must be the case that $F$ is not open. despite thinking on how ...
1
vote
1answer
34 views

Are the two formulations of Baire category theorem equivalent for arbitrary metric spaces?

As we all know, Baire Category theorem has two equivalent forms $X$ is a complete metric space, then the countable intersection of dense open sets is nonempty. $X$ is a complete metric space, $X$ is ...
1
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1answer
29 views

Ring structure on power set of a set.

Let $A$ be a non empty set. Let $P(A)$ denote the power set of $A$. $P(A)$ can be given a group structure in multiple ways. 1. Using Disjoint union a group operation Source may be worth noting in ...