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
2answers
53 views

Does a proper map have to be continuous?

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...
0
votes
0answers
48 views

Are Prevarieties irreducible?

In Goertz-Wedhorn, a prevariety is defined to be a connected space with functions that locally is an affine variety (were an affine variety is a space with functions that is isomorphic to the space ...
3
votes
4answers
111 views

How to show that $S^1$ with one point removed is still connected?

$S^1:= \{x \in \mathbb{R^2}: \|x\| = 1 \}$ Suppose $y_0 \in S^1$. Prove $(S^1-\{y_0\}, \mathcal{T}_{S^1- \{y_0\}})$ is connected. where $\mathcal{T}_{S^1- \{y_0\}}$ is the subspace topology coming ...
2
votes
2answers
148 views

Is a continuous image of $S^1$ to Hausdorff space locally connected?

Is a continuous image of $S^1$ to Hausdorff space locally connected? How do you prove this?
0
votes
1answer
106 views

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable? What I've tried: I list these facts: 1 A space $X$ is a Moore space iff $X$ is a $\sigma$-space and a $p$-space. 2 If $X$ is a ...
3
votes
3answers
291 views

Why are clopen sets a union of connected components?

The wikipedia page on clopen sets says "Any clopen set is a union of (possibly infinitely many) connected components." I thought any topological space is the union of its connected components? Why ...
0
votes
1answer
40 views

basis for topological spaces…

When we say that the sets $(a,b)$ form a base for usual topology on $\mathbb R$ why do we say in that context that $a,b$ are rational? Why not irrrational? And if we take only open intervals then ...
1
vote
1answer
86 views

Example of a homeomorphism on the real line?

I'm to give a short presentation on "basic topology" for a first semester undergrad analysis course. Naturally the professor does not expect me to master the topic, so I'm just trying to get some of ...
1
vote
2answers
53 views

One-to-one continuous mapping preserve openness?

I am reading a proof, and I see the following steps. Let $U\subset\mathbb R^n$ be open, and $g:U\to\mathbb R^n$ be one-to-one continuously differentiable where $det g'(x) \neq 0$ for all $x\in U$. ...
20
votes
7answers
2k views

Perfect set without rationals

Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals. (Or prove that it does not exist).
4
votes
4answers
362 views

Why do Topologies get “finer”?

Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that ...
2
votes
2answers
111 views

Why does the antipodal map on $S^2$ have degree $-1$?

I'm reading the post here by Arthur to explain that there is no smooth vector field on $S^2$. I don't understand it very well: The simplest I can remember off the top of my head is this: ...
0
votes
0answers
20 views

What quantities does a local topological region have in 3D?

If we take an infinite solid R3 and cut out a torus and sew it back in with Dehn surgery. This will create a local topological region in R3. I was thinking.. are there any characteristic values ...
61
votes
2answers
2k views

Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
2
votes
0answers
77 views

Conditions for $X\times Y=X\oplus Y$ holds? [closed]

In case $X=\mathbb R$ and $Y=\mathbb R$ the product $X\times Y=\mathbb R^2$ is plane. And, also $X\oplus Y=\mathbb R^2$. In this case $X\times Y=X\oplus Y=\mathbb R^2$. Does it always true? If not ...
0
votes
1answer
34 views

Mistake in Gabriel-Zisman regarding change-of-base of topological spaces?

In III.2.2 of Gabriel-Zisman, a Proposition is asserted which says that the base of change functor sending $X \to B$ to $X \times_{B} B'$, for any $B' \to B$ commutes with colimits in the $X$ ...
2
votes
0answers
92 views

Deformation retract of a triangle

Let $X \subset \mathbb{R}^2$ be a triangle equipped with the topology induced by the euclidean topology on $\mathbb{R}^2$ and let $Y \subset X$ be the subset made of two sides of the triangle. I need ...
3
votes
0answers
44 views

Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?). What ...
1
vote
1answer
38 views

How do I prove that $R^n\setminus R^k$ is homeomorphic to $S^{n-k-1}\times R^{k+1}$?

Let $k,n$ be positive integers such that $k<n$. How do I prove that $\mathbb{R}^n\setminus \mathbb{R}^k$ is homeomorphic to $S^{n-k-1}\times \mathbb{R}^{k+1}$? I tried to put specific integers in ...
8
votes
1answer
454 views

How to determine $\Omega(T)$?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and let $T\colon X\to X$ describe the following dynamics: 1 becomes a 2, 2 becomes a 0 and 0 becomes a 1 if at least one of its two neighbours is 1, ...
1
vote
0answers
48 views

Why (or when) is the direct limit of compact spaces paracompact?

I'm working through Milnor and Stasheff's Characteristic Classes and got stuck in chapter 5, p.66, where some (supposedly) easy facts about paracompact spaces are assembled. One of these is: ...
4
votes
2answers
62 views

Do “point of accumulation” and “boundary point” mean the same thing?

In my text it says, if a set $\Omega$ contains all points of accumulation $\{c\}$, then $\Omega$ is closed. I was surprised because people usually use "boundary point" in this context. And further ...
5
votes
5answers
3k views

Simpler definition of manifold

I'm new to topology but I must understand how it works to progress in my research. First, can anybody point me to a document that introduces topology in a "gentle" way. What pre-requisites do I need ...
4
votes
1answer
661 views

When is the closure of a path connected set also path connected?

What are the most general criteria we can impose on a locally path connected Hausdorff space $X$ and a path connected subset $A$ such that $\overline{A}$ is path connected? Do more restrictions need ...
2
votes
1answer
40 views

When is an ordered space scattered?

There is a concept of scattered in both order theory and topology. A topological space $X$ is scattered if every nonempty subspace has an isolated point. A linearly ordered set $( X , < )$ is ...
22
votes
2answers
254 views

Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ...
0
votes
2answers
42 views

Having difficulties with example 6, pg. 143, in Munkres' Topology.

I don't know how to prove that $A$ has no limit points. I maybe have to prove it by showing that for any point $(x, n), x \not= 1/n$ in $X$, one of its neighborhood does not intersect $A$, but I'm ...
2
votes
3answers
28 views

The union of open discs $C_n$ in $\mathbb{R}^2$ centered at $(n,0)$ with radius $n$

For each $n\ge 1$, let $C_{n}$ be the open disc in $\mathbb{R}^2$, with centre at the point $(n,0)$ and radius equal to $n$. Then $\mathcal{C} = \cup C_{n}$ is ...
4
votes
1answer
436 views

Lindelöf's Covering Theorem

If $A \in \mathbb R^n$ and $F$ be an open covering of $A$. Then there is a countable subcollection of $F$ which also covers $A$. Proof: Let $G=\{A_1,A_2, \cdots\}$ denote the countable collection of ...
14
votes
2answers
179 views

Geometric reason as to why $H^2$ of the Klein bottle is $\mathbb{Z}/2\mathbb{Z}$?

I was reading this document when I came across the following: Recall that $H^2(K; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. Here $K$ denotes the Klein bottle. Is there a good geometric ...
3
votes
0answers
56 views

Contractible pieces of $GL(n,\mathbb{C})$

Is $GL(n,\mathbb{C})$ contractible for any $n$? My intuition is telling me it is not, because the determinant maps the general linear to $\mathbb{C}\setminus 0$ which is not contractible. If there ...
2
votes
1answer
73 views

Hierarchy of Mathematical Spaces

I really got lost among all those many different spaces in mathematics, and I got really confused what is special case of what. For example, I knew for long time vector spaces, then Hilbert spaces, ...
1
vote
1answer
28 views

Separable iff Lindelöf for pseudometric spaces

I'm trying to prove, for $X$ a pseudometric space $$X \text{ Lindelöf } \Leftrightarrow X \text{ separable }$$ Here are some of my ideas so far - the forward direction should work: $(\Rightarrow)$ ...
0
votes
0answers
27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
-2
votes
1answer
37 views

Is the boundary of a compact connected subset of $\mathbb R^n, n>1$ connected? [closed]

Let $A\subset \mathbb R^n$ be compact and connected, where $n\ge2$. Is the boundary $\partial A$ connected?
0
votes
2answers
78 views

When $X\times (Y\times Z)=(X\times Y)\times Z$ in product topology?

Under what conditions $X, Y, Z$ must have so $X\times (Y\times Z)=(X\times Y)\times Z$? and proof? $X, Y, Z$ are topological spaces and $\times$ represents product topology.
-5
votes
2answers
116 views

A separable paracompact space is Lindelöf [closed]

Show that: A separable paracompact space is Lindelöf. Thanks for your help.
2
votes
0answers
26 views

Topological reflection of pretopological closure operator

Given a pretopological space $(X,\mbox{cl})$ where $\mbox{cl}$ is a pretopological closure operator. How does one find the topological reflection of $(X,\mbox{cl})$? I know of a way namely by ...
2
votes
2answers
55 views

Need example of: Algebraic sum of closed vector subspaces need not be closed

I've read somewhere that given two closed subspaces $V_1,V_2$ in topological vector space $X$, their algebraic span $V_1+V_2=\{x_1+x_2 |x_i \in V_i, i=1,2\}$ need not be closed. I always thought that ...
2
votes
1answer
32 views

Hausdorff Lindelöf Space is Regular?

I think we can use same argument for saying regular Lindelöf space is normal to prove Hausdorff Lindelöf space is regular. But, I didn't heard about this proposition. What is the problem of using same ...
6
votes
2answers
70 views

$A \subseteq \mathbb{R}^n$ closed and connected. Prove $\{x \in \mathbb{R}^n \mid d(x, A) \le \varepsilon\}$ is path-connected

I've encountered the following question: Say $A \subseteq \mathbb{R}^n$ is a closed and connected set. Prove $\{x \in \mathbb{R}^n \mid d(x, A) \le \varepsilon\}$ is path-connected. I'm not really ...
0
votes
1answer
177 views

Show that $f(\bar A) \subset \overline{f(A)}$.

Let $X$ be a metric space, and $Y\subset X$ a subset. A point $x\in X$ is adherent to $Y$ if $B(x;r) \cap Y \neq \emptyset$ $\forall r > 0.$ The closure of $Y$ is then defined as $\bar Y := \{x\in ...
2
votes
0answers
47 views

Density of subset with nonlocal boundary condition

I am having difficulty proving that $E=\bigcap_{n\geq 0} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a dense subset of: $F=\{f\in C^2 (\mathbb{R}) : ...
-1
votes
2answers
89 views

Is every Hausdorff space metric? [duplicate]

My question is very simple. I know every metric space is Hausdorff, but the converse is true? anyone knows some counterexample? Thanks
0
votes
2answers
20 views

The Cauchy-Schwarz inequality for Hermitian forms

Let $V$ be a vector space over a field $\mathbb K$ and $f$ be a nonnegative Hermitian form on $V$. Then, $\forall x,y \in V$: $$|f(x,y)|^2 \le f(x,x)f(y,y)$$ Here's one proof: For an ...
1
vote
1answer
34 views

The boundary of the intersection of a decreasing sequence of convex sets

Let $\Omega_1 \supset \Omega_2 \supset \dots$ be a sequence of bounded, open and convex sets in $R^n$ with $\Omega :=\operatorname{int}(\overline{\bigcap \Omega_n})$ nonempty. It seems that $\partial ...
4
votes
1answer
53 views

Surjectivity of $\mathcal{id}_{\mathbb{R}^n}+g$ when $g$ is a contraction?

Assume $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is a contraction and consider $h=\mathcal{id}_{\mathbb{R}^n}+g$. The map $h$ is injective. Is it always surjective? My question has the following ...
3
votes
1answer
43 views

Prove that if the closure of each open ball in compact metric space is the closed ball with the same radius, then any ball in this space is connected

I'm having some difficulty with the following problem in general topology: Prove that if the closure of each open ball in compact metric space is the closed ball with the same center and radius, then ...
0
votes
0answers
26 views

Is the subspace generated by complete othogonal subspaces closed?

$E$ is a vectorial space equipped with an inner product $\langle \cdot, \cdot \rangle$. $(E_i)_{i \in I}$ is a family of complete pairwise orthogonal subspaces. Is the subspace $V$ generated by the ...
2
votes
1answer
58 views

What are the Results of the First and Second Axioms of Countability?

What are the consequences of a space being first or second countable? What was the motivation for these axioms in the first place?