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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3
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1answer
97 views

If a set could be represented as “arbitrary fine” finite union of open balls, then it is not closed

If $V$ is a subset of a metric space, such that for every $\varepsilon > 0$ there exists a finite number of open balls $B_{\varepsilon}(x_i)$ such that $$ V = \bigcup_{i = 1}^n ...
3
votes
3answers
113 views

If the closure operator interchanges with tacking finite intersections, is this then true for a countable intersection?

Let $(A_{i})_{i\in \mathbb{N}}$ be a family of subsets of $(X,\tau)$ satisfying $\overline{\bigcap_{i=1}^{n} A_{i}} = \bigcap_{i=1}^{n} \overline{A_{i}}$ for any $n \in \mathbb{N}$. Is it true (or ...
3
votes
3answers
366 views

Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
3
votes
3answers
484 views

Intersection of a closed set and compact set is compact [duplicate]

I've been stuck on the following problem for several days: Let $(M,d)$ be an arbitrary metric space and $S, T$ be subsets of $M$. If $S$ is closed and $T$ is compact, then $S \cap T$ is compact. I ...
3
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1answer
55 views

On the proof of deformation lemma “boundedness”

Book- Evans partial differential equation. In the proof of deformation lemma how to say that $V(u)=-g(u)h(\lVert I'(u)\rVert)I'(u)$ is bounded. And how to say that the mapping $u \to ...
3
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1answer
129 views

What do we mean when we say a differential form “descends to the quotient”?

Let $S$ be a surface and let $f:S\rightarrow S$ be a diffeomorphism. We define the mapping torus $M_f$ of the pair $(S,f)$ to be the quotient $$(S\times I) /\sim \quad \text{ where } \ (1,x) \sim ...
3
votes
1answer
851 views

Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...
3
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1answer
286 views

Partial Order induced topology

I was wondering if there is a canonical topology induced by a partial order on a set and how that relates to the total ordering topology (if it can be extended to a total ordering). I thought maybe ...
3
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2answers
117 views

Can a set of non self-intersection points of a space-filling curve contain an arc?

Consider a continuous surjection $f:[0,1]\to[0,1]\times[0,1]$. It can be proved that set of self-intersection points must be dense. In the Hilbert curve, the set of self-intersections are points ...
3
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1answer
55 views

Is it possible that the union of a Bernstein set and a singleton isn't a Bernstein set?

Since the construction of a partition of two Bernstein sets is almost identical to that of a partition of three in an uncountable Polish space. It's possible that the union of a Bernstein set and a ...
3
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0answers
254 views

Euler Characteristic of surface with boundary puncture.

I am studying a course on differential geometry. I saw the formula for the euler characteristic of a surface with $g$ holes and $b$ boundaries components and $n$ punctures is $2-2g -b +n$. In ...
3
votes
1answer
200 views

Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
3
votes
1answer
79 views

How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
3
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2answers
227 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
3
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3answers
302 views

If $X$ is compact and $f:X\to\mathbb{R}$ is continuous, then $f$ attains the values $\inf\{f(x):x\in X\}$ and $\sup\{f(x):x\in X\}$

I need to show that if $f$ is continuous real valued function on the compact space $X$, then there exist points $x_1,x_2\in X$ such that $f(x_1)=\inf\{f(x):x\in X\}$ and $f(x_2)=\sup\{f(x):x\in X\}$. ...
3
votes
1answer
55 views

The weight of Sorgenfrey line

What is the weight of Sorgenfrey line $S$? Weight $(X)=\min\{|\mathcal{B}|: \text{ $\mathcal{B}$ a base for $X$}$} + $\omega$ Thanks ahead.
3
votes
2answers
258 views

Picturing the discrete metric?

A chapter on topology talks about this and asks me to "think of a model for the discrete metric on M" where M may have 1, 2, 3, 4 points. Here is what I think: 2 points = 2 distinct points 3 ...
3
votes
1answer
149 views

Let $X = [0,2]$ and $A = \{0,1,2\}$. Prove that $X / A$ is homeomorphic to $C_{1}$ ∪ $C_{-1}$

Let $X = [0,2]$ and $A = \{0,1,2\}$. Prove that $X / A$ is homeomorphic to $C_{1}$ ∪ $C_{-1}$, where $C_1$ and $C_{-1}$ are the circles of radius $1$ centered at $(1, 0)$ and $(-1, 0)$ , respectively. ...
3
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1answer
39 views

Lindelöfness in $\omega_1$

Is every Lindelöf subspace of the ordinal space $\omega_1$ countable?
3
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1answer
450 views

A locally compact subset of a locally compact Hausdorff space is locally closed

Let $X$ be a locally compact and Hausdorff space. Show that if $Y \subset X$ is locally compact, then $Y$ is locally closed, in essence $Y$ is an open subset of $\overline{Y}$, where $\overline{Y}$ ...
3
votes
1answer
154 views

Close linearly independent sequence

X is an infinite-dimensional normed space. Show that there is a linearly independent sequence ${x_n}$ in X, such that for any sequence ${\epsilon_n} > 0 $ for all n, there is a sequence ${y_n}$ ...
3
votes
1answer
266 views

Is the $\sigma$-algebra generated by a product topology a subset of the product $\sigma$-algebra of the individual topologies?

Is the generated $\sigma$-algebra of a product topology a subset of the product $\sigma$-algebra of the individual topologies? Formally, let $\Theta$ be some non-empty set (to serve as a set of ...
3
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0answers
198 views

A question from Arhangel'skii-Buzyakova

Recently, I am reading the paper: On linearly Lindelöf and strongly discretely Lindelöf spaces by Arhangel'skii and Buzyakova. Here is the Lemma 2.2 in paper. (Sorry for the picture is not clear.) ...
3
votes
1answer
397 views

A closed set in a metric space is $G_\delta$

How do I prove that a closed set $F$ in the metric $(X,d)$ is $G_\delta$. Let $n\in \mathbb{N}$. I consider $B_n={F}=\bigcup_ {x\in F} B(x,{1\over n})$, which is a collection of an open ball. Then ...
3
votes
1answer
685 views

Klein-bottle and Möbius-strip together with a homeomorphism

Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement: The Klein Bottle is homeomorphic to the union of two copies of a Möbius ...
3
votes
1answer
769 views

Proving that all polynomials are continuous

Prove that all polynomials from $\mathbb{R}$ to $\mathbb{R}$ are continuous. Now this is from a topological point of view. I thought that maybe induction would work here? Initial Case: $f(x) = ...
3
votes
1answer
305 views

Topology - axioms of metric space - convergency - Cauchy

Let $(X,d)$ be a complete metric space and $U \subseteq X$, $U \neq X$, its open subset. Define a function $\rho\colon U \times U \rightarrow [0, \infty)$ as: ...
3
votes
4answers
129 views

What does continuity of inclusion means?

If $A,B,C$ are three spaces such that $A\subset B\subset C$ and $A$ is dense in $C$. Now my teacher said that the inclusion between three spaces are continuous and so you can directly say that $B$ is ...
3
votes
1answer
226 views

Let G be a bipartite graph all of whose vertices have the same degree d. Show that there are at least d distinct perfect matchings in G

Let G be a bipartite graph all of whose vertices have the same degree d. Show that there are at least d distinct perfect matchings in G. (Two perfect matchings M1 and M2 are distinct if M1 does not ...
3
votes
3answers
459 views

Closures and Interiors in a topology

Let $X$ be a set. Let $A$ be a proper non-empty subset of $X$. Let $\tau = \emptyset \cup \{U \in P(X): A \subseteq U\}$ Question: Find the interior and closure of $A$ and prove that they are ...
3
votes
1answer
101 views

Help needed with last step in proof of Tychonoff theorem

I am going through the proof in this book of Tychonoff theorem. First I understand proof of Alexander subbase theorem. Then the proof of Tychonoff theorem uses that. However, I am unable to follow ...
3
votes
2answers
51 views

Is it true that, $A,B\subset X$ are completely seprated iff their closures are?

If $A,B\subset X$ and $\overline{A}, \overline{B}$ are completely seprated, so also are $A,B$. since $A\subset \overline{A}$, $B\subset \overline{B}$ then, $f(A)\subset f(\overline{A})=0$ and ...
3
votes
0answers
150 views

Moscow space-Examples

A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G‎_{δ}$‎‎‎ ‎-subsets of $X$ . For example, Every first countable $T_1$ ...
3
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1answer
80 views

Proving that a $T_{1\frac{1}{3}}$-space is $T_1$-space

Call a space $T_{1\frac{1}{3}}$ if every sequence in it has at most one limit. A $T_1$ space is a space in which for two distinct points $a$ and $b$, there are open sets $U$ and $V$ for which $a$ ...
3
votes
2answers
134 views

Does $d(x+u, y + v) \le d(x, y) + d(u,v)$ holds for every metric?

The title said it, I want to prove that $$ d(x+u, y + v) \le d(x, y) + d(u,v) $$ for every metric $d$. If the metric is induced by a norm, i.e. $d(x,y) := ||x-y||$, then this is easy. \begin{align*} ...
3
votes
1answer
453 views

Problem 4 chapter 2: functional analysis (Rudin)

$L^1$, $L^2$: usual Lebesgue spaces on the unit interval. Show that $L^2$ is of the first category (meager) in $L^1$, in three ways: (a) Show that $F_n:=\{f:\int|f|^2 \leq n\}$ is closed in ...
3
votes
1answer
177 views

Why this space is homeomorphic to the plane?

I'm trying to see why this picture below is homeomorphic to the $\mathbb R^2$. It's really hard, please I need an intuitive idea of this. This seems very weird for me, I need help. Thanks a lot
3
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1answer
141 views

The unit square stays path-connected when you delete a cycle-free countable family of open segments?

This question was inspired to me by Lukas Geyer’s recent question. A positive answer to this question would easily entail a positive answer to Lukas’ question also, and a negative answer would ...
3
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2answers
235 views

Continuous functions on discrete product topology

Let $A = \{a_1,\dots,a_m\}$ be a finite set endowed with a discrete topology and let $X = A^{\Bbb N}$ be the product topological space. I wonder which bounded functions $f:X\to\Bbb R$ are continuous ...
3
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1answer
68 views

$(a,b)$ homemorphic to $(c,d) \cup (h,k)$

Given $(a,b)$, and $(c,d) \cup (h,k)$ subspaces of $\mathbb{R}$ with the absolute value metric induced topology where $c<d<h<k$ are any real numbers. We have just started working with ...
3
votes
2answers
139 views

Must $f$ be continuous?

Let $f:\mathbb{R^n}\to\mathbb{R^m}$ be a function such that the image of any closed bounded set is closed and bounded. Must $f$ be continuous?
3
votes
4answers
861 views

Can a graph be non 3-colourable without having k4 as a sub graph?

As the question asks, is it possible for a graph to have a chromatic number larger than three without it having a 4 vertice complete graph as a sub-graph?
3
votes
2answers
129 views

Sequential space implies countable tightness?

Recently, I met across a question, i.e., Sequential space implies countable tightness? sequential space = $X$ is sequential if $A \subset X$ and $A$ is not closed implies that there is a sequence ...
3
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2answers
227 views

Prove integral metric is separable

I have no idea how to prove that the space $X$ of all integrable functions on the interval $[0,1]$, for $f,g\in X$, with the following metric: $$\rho(f,g)=\int|f(x)−g(x)|~dx$$ is separable. I'll ...
3
votes
4answers
225 views

A question about a closed set

Let $X = C([0; 1])$. For all $f, g \in X$, we define the metric $d$ by $d(f; g) = \sup_x |f(x) - g(x)|$. Show that $S := \{ f\in X : f(0) = 0 \}$ is closed in $(X; d)$. I am trying to show that $X ...
3
votes
1answer
354 views

Help proving the primitive roots of unity are dense in the unit circle.

I'm having difficulty understanding how to prove that the primitive roots of unity are in fact dense on the unit circle. I have the following so far: The unit circle can be written ...
3
votes
2answers
113 views

map from $\mathbb{C}$ to $\mathbb{C}/L$ is open map?

Let $w_1,w_2\in\mathbb{C}$ be linearly independent vectors and let$$L=\{m_1w_1+m_2w_2:m_1,m_2\in\mathbb{Z}.\}$$ How does one show that the projection map $\pi:\mathbb{C}\rightarrow\mathbb{C}/L$ is ...
3
votes
5answers
1k views

$f\colon M\to N$ continuous iff $f(\overline{X})\subset\overline{f(X)}$ [duplicate]

Possible Duplicate: Continuity and Closure $f\colon M\to N$ is continuous iff for all $X\subset M$ we have that $f\left(\overline{X}\right)\subset\overline{f(X)}$. I only proved ...
3
votes
1answer
297 views

Extreme boundary of a compact, convex, metrizable set is $G_\delta$

Let $X$ be a topological vector space (no assumptions about local convexity are made in the question, though I am worried they might be required). Suppose $K\subset X$ is a compact, convex, metrizable ...
3
votes
3answers
221 views

Discontinuous function sending compacts to compacts

I know that the condition that $f(X)$ is compact if $X$ is compact should not be sufficient to say that $f$ is continuous, but I can't come up with an example of such discontinuous $f$. What is it? ...