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; ...

learn more… | top users | synonyms (3)

1
vote
3answers
93 views

$X$ is an Alexandroff space iff each point in $X$ has a minimal open neighborhood.

$X$ is an Alexandroff space iff each point in $X$ has a minimal open neighborhood.
1
vote
1answer
30 views

Help proving a set is not compact (without Heine-Borel Theorem).

Show directly from the definition (that is without using the Heine-Borel theorem that the set $\{(x,y) \in R^2:x^2+y^2 \ge \}$ is not a compact subset of $R^2$. I'm not sure where to start. Any help ...
1
vote
1answer
64 views

How are linked rings homeomorphic to seperated links?

I'm currently reading "Geometry, Topology and Physics" by Mikio Nakahara. In his book there the following exercise: Show that two figure in figure 2.109(b) [see below] are homeomorphic to each ...
1
vote
1answer
35 views

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and vice versa

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and ...
1
vote
1answer
61 views

What does $X^Y$ mean?

I have been reading about the Stone-Čech compactification recently and one way of constructing it is by considering the map $h:X\rightarrow I^{C}:x\mapsto(fx)_{f\in C}$ where $I$ is the closed unit ...
1
vote
1answer
109 views

Quotient Map vs Embedding (Topology)

Problem 1: Can any quotient $\tilde{X}$ of $X$ be embedded in $X$? Moreover, does any (surjective) quotient map $\pi:X\to\tilde{X}$ left split with an (injective) embedding $\iota:\tilde{X}\to X$? ...
1
vote
1answer
153 views

meaning of quotient topology in munkres book

I can't understand quotient topology in Munkres' book it says at page 137 :$p$ from $X$ to $Y$ a quotient map is equivalent to saying that $p$ is continuous & $p$ maps saturated open sets of ...
1
vote
1answer
33 views

Compact sets problem

Find two compact subsets $A,B\subseteq\mathbb{R^2}$ such that $A\times[0,1]$ is homeomorphic to $B\times [0,1]$, but $A$ is not homeomorphic to $B$. I'm really not sure where to go with this one. Any ...
1
vote
1answer
60 views

Showing the operation of taking the Composition of Paths is Continuous

Let $I=[0,1]$. Let $X$ be a topological space, and denote by $C(I,X)$ the space of all paths in $X$, equipped with the compact-open topology. We denote by $P_{x,y}(X)$ the fiber of $(x,y)$ along the ...
1
vote
2answers
289 views

A contractible space is path connected.

Let the space be $X$ and $\rm{id} \simeq x_0$ where $\rm{id}$ is the identity map on $X$ and $\rm x_0$ is some fixed point in $X$. How do I show that for any two points $a,b \in X$ there is a ...
1
vote
2answers
86 views

Uniqueness of Topology and Basis

In measure theory, we know there is a (unique) minimal $\sigma$-algebra generated by a generator. I am wondering whether this applies to topology and its basis. There are two directions to consider ...
1
vote
1answer
34 views

Can anything be said for the topology of a topological monoid?

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$. They are uniformisable and hence are ...
1
vote
1answer
146 views

What is the definition of a product topology?

I am new to advanced mathematics and I recently started reading a book on topology. I am struggling to understand what it is saying in this paragraph. This is what it says: Let $E_i$ ...
1
vote
1answer
114 views

Question on Baire Property

In reading Banach's book, Theory of Linear Operations, I have a question on the definition of Baire property, or Baire condition in Banach's book. Here is the definition in Banach's book: Definition. ...
1
vote
1answer
84 views

Intermediate Value theorem, $nth$ root function and continuity

So this problem is.. ridiculous to be honest. I have no idea where to start or what to do. Any help is appreciated. For the record, I am using the metric spaces definition of continuity.
1
vote
2answers
70 views

Every function from a discrete subset is continuous

Let $D \subset \mathbb{C}$ be a discrete subset and let $f : D \mapsto \mathbb{C}$ be a function. Show that $f$ is continuous. What's the best way to do this? I was thinking a proof by contradiction ...
1
vote
2answers
103 views

Let $\{A_α\}$ be a collection of connected subspaces of $X$; let A be con. sub. of X. Show that if $A∩A_α≠ ∅$ $∀α$, then $A∪(∪ A_α)$ is connected.

Let $\{A_\alpha\}$ be a collection of connected subspaces of $X$; let $A$ be connectted subspace of $X$. Show that if $A\cap A_\alpha \neq \emptyset$ for all $\alpha$, then $A\cup(\cup ...
1
vote
1answer
30 views

“the standard two-fold branched cover of $CP^2$”

What could the following sentence mean: $\iota : S^2\times S^2 \rightarrow \mathbb{C}P^2$ is the standard two-fold branched cover, branched along the diagonal. What I can think of is to ...
1
vote
1answer
82 views

Show that $T_\infty=\{U : X-U \text{ is infinite or empty or all of $X$} \}$ is a topology on $X$ [duplicate]

Let $X$ be a set. Show that the collection $$T_\infty=\{U : X-U \text{ is infinite or empty or all of $X$} \}$$ is a topology on $X$. Well $X-X=\emptyset$, so $X \in T_\infty$. And $X-\emptyset= X$ ...
1
vote
1answer
529 views

Topology - interval homeomorphic to another interval

{a.} Prove that any open interval $(a, b)$ is homeomorphic to the interval $(0, 1)$. Define $f:(a, b) \to (0, 1)$ by $f(x)=(x-a)/(b-a)$, which is one-to-one and onto. Consider $f^{-1}:(0, 1) \to (a, ...
1
vote
2answers
56 views

Question about linear operators and neighborhoods

Suppose $X,Y$ are normed spaces and $T: D(X) \to Y $ is a linear operator. Suppose $V$ is a neighborhood of $T\Theta = \Theta $(zero vector). Can we say that $U = V + Tx_0$ is a neighborhood of $x_0$ ...
1
vote
1answer
44 views

topology space $T_4$ question

Prove that A topology space $X$ is $T_4$ if and only if for all disjoint closed subsets $F$ and $F$ of $X$ there exist open subset $U$ and $V$ of $X$ such that $F \subset U$ , $F' \subset V$, and ...
1
vote
3answers
205 views

The continuous image of a sequentially compact set is also sequentially compact.

Let $S$ be a sequentially compact set and let $f : S\to R$ be continuous. Then the image $f(S)$ is sequentially compact.
1
vote
1answer
139 views

A dense subalgebra of $C(X)$ that separates points

Any idea how to do this problem: If $X$ is a compact Hausdorff space and $A$ a subalgebra of $C(X)$ , where $C(X)$ is the algebra of all continuous functions, such that $A$ contains the constant ...
1
vote
1answer
63 views

Show that points on a smooth manifold can be separated by smooth function

I need to show that given $x,y\in M, x\neq y$ where $M$ is a smooth manifold, $\exists$ a smooth function $f:M\longrightarrow \mathbb{R}$ such that $f(x)=0, f(y)=1.$ Attempt of the proof: Since $M ...
1
vote
1answer
88 views

A space $Y$ is Hausdorff iff the diagonal $D = \{(y_1,y_2)\in Y\times Y: y_1=y_2\}$ is closed [duplicate]

Show that the topological space $Y$ is Hausdorff iff the diagonal $D = \{(y_1,y_2)\in Y\times Y: y_1=y_2\}$ is a closed subset of $Y\times Y$. My attempt: First, suppose $Y$ is Hausdorff. Then ...
1
vote
1answer
61 views

Why is a quotient mapping not necessarily open? [duplicate]

I read that a quotient mapping is not necessarily open. I wonder why that is. Say we have a quotient mapping $f$ between $(X,\mathfrak{A})$ and $(Y,\mathfrak{B})$. Let us take an open set $O\subset ...
1
vote
1answer
42 views

Can a finite subbasis create an infinite topology / topological space?

I've translated the following from my German textbook, so please correct me if there is something wrong or strange. Definitions Let $X$ be a set and $\mathfrak{T} \subseteq \mathcal{P}(X)$ with fits ...
1
vote
2answers
38 views

What does metrizablity means

Let $(X,T)$ be a topological space, and let $f$ be a homeomorphism form $(X,T)$ to any metric space. Than we say that $(X,T)$ is metrizabile. This means that X can be equipped with a metric and can be ...
1
vote
1answer
114 views

Showing a projection map on restricted to a subset is not an open map

I'm working on a problem from Munkres about open and closed maps. Here's the problem: "Let $\pi_1 : \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ be projection onto the first coordinate. Let ...
1
vote
1answer
55 views

Continuos function between topological space and metric space imply metrizality of topological space?

Im trying to prove that if the conclusion of urysohn's lemma holds then X is normal. I was thinking that if X is metrizable then X i T4 then X is normal. But dont feel 100 percent confident with my ...
1
vote
2answers
37 views

$\Omega$ open and $K$ compact $\Rightarrow B_\varepsilon(0)+K\subset\Omega$ for $\varepsilon$ small enough.

Let $\Omega\subset\mathbb{R}^N$ be an open bounded set and $K\subset\Omega$ compact. I'm trying to prove that for $\varepsilon$ small enough we have $(B_\varepsilon(0)+K)\subset\Omega$. Consider the ...
1
vote
1answer
33 views

Theorem why are $X_n$ closed?

Our professor gave us the following theorem. Let $T$ be a compact topological space then any infinite subset of T has at least a limit point. Proof $T$-compact space. Suppose $B$ subset of $T$ with ...
1
vote
1answer
317 views

Möbius Strip Cylinder

I am having trouble seeing why for the the regular cylinder $C=\{(x,y,z)|x^2+y^2=1,|z|\le 1\}$, $C/\mathbb{Z}_2$ is homeomorphic to the Möbius band ($x_0 \in C$). Can someone explain?
1
vote
2answers
62 views

Is the function that gives you the measure of the neighborhood Borel?

Let $X$ be a compact metric space (with $\epsilon -$balls $B_{\epsilon }$) and $\mu $ a Borel probability measure. Let $a,\epsilon >0.$ Is the set $\left\{ x\in X:\mu (B_{\epsilon }(x))\geq ...
1
vote
1answer
44 views

Extension of funcion

I think its right but Im not sure. I have topological space (exactly manifold - second countable, Hausdorff, local Euclidean topological space) M, dim M=m. Let $A \subset M$ is closed set, dim A=n, ...
1
vote
2answers
34 views

question about continuity of a function that maps a non-compact set to the plane

Let $f: [0, 2 \pi) \to \mathbb{R}^2 $ be function with $$ f(x) = ( \cos x , \sin x ) $$ Notice $f( [0, 2 \pi ) ) = S^1 $. I want to know about the continuity of the function $$ f^{-1} : S^1 \to ...
1
vote
2answers
82 views

How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
1
vote
1answer
69 views

Connectedness and Continuity

Suppose $S^n\subset \mathbb{R}^n$ be the set of all points in $\mathbb{R}^n$ with unit Euclidean norm. That is $$S^n=\{x\in\mathbb{R}^n~:~\|x\|=1\}$$ If $f:S^{n-1}\rightarrow\mathbb{R},~n>2$, is ...
1
vote
1answer
39 views

How is the identity mapping from $\Bbb{R_\ell}\to\Bbb{R}$ continuous?

Munkres' book on Topology (Pg 122 of the Second edition) says The identity mapping $f:\Bbb{R_\ell}\to\Bbb{R}$ defined by $f(x)=x$ (identity mapping) is continuous I don't understand how. An ...
1
vote
1answer
42 views

Determining if this collection of sets is a topology on the real numbers?

Let $\tau$ consist of $\mathbb{R}$, $\emptyset$ and every interval $[-r, r]$, for $r$ any positive irrational number. i) $\mathbb{R}, \emptyset$ are in $\tau$ ii) $[-r_1, r_1] \bigcup [-r_2, r_2] ...
1
vote
1answer
52 views

The set of continuous functions is closed

Let $X$ be the set of bounded functions on a closed interval $[a,b]$. Note that $d(f,g) = \sup|f(t) − g(t)|$. If $A\subset X$ is the set of continuous functions on $[a, b]$, show that $A$ is closed in ...
1
vote
1answer
41 views

Is the “vanishing set” the same as the “kernel” of a function?

I've just read of a set called "vanishing set" in my topology lecture notes. It seems to be the kernel of a special type of functions. Definitions The lecture notes are in German, but I try to ...
1
vote
1answer
122 views

What are the typical approaches to showing that some function sequence does not converge uniformly?

The following problem is from Munkres's Topology (Exercise 6 of Section 21 "The Metric Topology (continued)", 2nd edition). Exercise: Define $f_n : [0,1] \to \mathbb{R}$ by the equation $f_n(x) = ...
1
vote
3answers
89 views

Compactness of subspaces of $\mathbb R^\infty$.

Let $\mathbb R^\infty=\bigcup \mathbb R^i$ (identifying $\mathbb R^i$ with subspace of $\mathbb R^{i+1}$) and $U\subset \mathbb R^\infty $ is open iff $U\cap \mathbb R^i$ is open in $\mathbb R^i$ for ...
1
vote
2answers
117 views

Elementary properties of closure

Hi everyone I'd like to know if the following is correct. I really appreciate any suggestion. (Honestly the only one that matters me is the second property the others are easy, I think) Thanks. ...
1
vote
1answer
112 views

Is the set of interval of the form $[a,b)$ in the set of real numbers a basis for some topology?

Prove that each of following are bases for topologies on the prescribed sets: the set of interval of the form $[a,b)$ in the set of real numbers. I think it is because Let $X=[a,b)$. This ...
1
vote
1answer
40 views

Minimizer and invariance of normal projection energy of a knot

While reading the paper, A simple energy function for knots, I understand that the authors have proved the two conditions of the first page for the normal projection energy of a knot. But I failed to ...
1
vote
1answer
60 views

Does homeomorphism preserve second countabity?

This seems obviously true and i proved it, but i couldn't find this in googls, so i'm asking this to make sure. Let $X,Y$ be topological spaces. Let $H:X\rightarrow Y$ be a homeomorphism. Is $Y$ ...
1
vote
1answer
37 views

How can a bounded subspace of the left order topology be compact?

I want to show that every bounded set equipped with the left order topology is compact. This is a statement I found on a wikipedia page and appearently it is lifted from the book Counterexamples in ...