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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25
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2answers
2k views

Set of continuity points of a real function

I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
22
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14answers
13k views

best book for topology?

I am a graduate student of math right now but i was not able to get a topology subject in my undergrad... i just would like to know if you guys know the best one..
16
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5answers
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).
9
votes
4answers
3k views

Projection map being a closed map

Let $\pi: X \times Y \to X$ be a projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is ...
47
votes
4answers
4k views

Continuous bijection from $(0,1)$ to $[0,1]$

Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map.
18
votes
7answers
4k views

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
12
votes
7answers
2k views

Choosing a text for a First Course in Topology

Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
2
votes
1answer
541 views

Why is this quotient space not Hausdorff?

I am trying to show that the following space is not Hausdorff. Consider the topological space $S^1$, and let $r$ be an irrational number. Consider the action of $\mathbb{Z}$ on $S^1$ given by $$ ...
8
votes
1answer
711 views

Polish Spaces and the Hilbert Cube

I've been trying to prove that every Polish Space is homeomorphic to a $G_\delta$ subspace of the Hilbert Cube. There is a hint saying that given a countable dense subset of the Polish space $\{x_n : ...
5
votes
4answers
2k views

$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed

Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$ Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$. ...
22
votes
1answer
1k views

What concept does an open set axiomatise?

In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ ...
16
votes
2answers
767 views

The set of ultrafilters on an infinite set

After recently learning about filters and ultrafilters, we looked into further problems and properties. I am having trouble with this one: If $X$ is an infinite set, then the set of all ultrafilters ...
10
votes
6answers
3k views

Functions which are Continuous, but not Bicontinuous

What are some examples of functions which are continuous, but whose inverse is not continuous? nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry.
24
votes
5answers
4k views

What's going on with “compact implies sequentially compact”?

I've seen both counterexamples and proofs to "compact implies sequentially compact", and I'm not sure what's going on. Apparently there are compact spaces which are not sequentially compact; quick ...
9
votes
1answer
950 views

Topologist's sine curve

Is there a (preferably elementary) proof that the graph of the function $y$ defined on $[0,1)$ by $$ y(x) = \left\{\begin{array}{ll} \sin\left(\frac{1}{x}\right) & \mbox{if $0\lt x \lt 1$,}\\ 0 ...
8
votes
5answers
747 views

Subgroup of $\mathbb{R}$ either dense or has a least positive element?

Let's say $G$ is some additive subgroup of $\mathbb{R}$ that has at least two elements. From what I understand, $G$ is then either dense in $\mathbb{R}$, or has some least positive element. What is ...
8
votes
5answers
2k views

Locally Constant Functions on Connected Spaces are Constant

I am trying to show that a function that is locally constant on a connected space is, in fact, constant. I have looked at this related question but my approach is a little different than the suggested ...
6
votes
3answers
572 views

Accumulation points of uncountable sets

Given any uncountable subset $S$ of the unit interval. Then $S$ clearly has an accumulation point and indeed uncountably many (which might also be a nice exercise). So my question is: Is there an ...
12
votes
7answers
4k views

Proving that the set of limit points of a set is closed

From Rudin's Principles of Mathematical Analysis (Chapter 2, Exercise 6) Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed. I think I got it but my argument is ...
1
vote
2answers
435 views

$\limsup $ and $\liminf$ of a sequence of subsets relative to a topology

From Wikipedia if $\{A_n\}$ is a sequence of subsets of a topological space $X$, then: $\limsup A_n$, which is also called the outer limit, consists of those elements which are limits of ...
63
votes
18answers
6k views

Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
55
votes
5answers
2k views

Defining a manifold without reference to the reals

The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. ...
14
votes
8answers
2k views

How to understand compactness? [duplicate]

How to understand the compactness in topology space in intuitive way?
6
votes
2answers
897 views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in ...
11
votes
2answers
4k views

A and B disjoint, A compact, and B closed implies there is positive distance between both sets

Claim: Let $X$ be a metric space. If $A,B\in X$ are disjoint, if A is compact, and if B is closed, then $\exists \delta>0: |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$. Proof. ...
16
votes
2answers
794 views

Any two points in a Stone space can be disconnected by clopen sets

Let $B$ be a Stone space (compact, Hausdorff, and totally disconnected). Then I am basically certain (because of Stone's representation theorem) that if $a, b \in B$ are two distinct points in $B$, ...
12
votes
3answers
1k views

Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?

This is an exercise from Kelley's book. Could someone help to show me a proof? It seems very natural, and it is easy to prove by utilizing the arctan function in $\mathbb{R}^1$. Thanks a lot.
8
votes
4answers
2k views

Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
6
votes
1answer
1k views

Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
7
votes
4answers
2k views

Equivalent metrics determine the same topology

Suppose that there are given two distance functions $d(x,y)$ and $d_1 (x,y)$ on the same space $S$. They are said to be equivalent if they determine the same open sets. Show that $d$ and $d_1$ are ...
8
votes
4answers
526 views

Are there more general spaces than Euclidean spaces to have the Heine–Borel property?

From Wikipedia A metric space (or topological vector space) is said to have the Heine–Borel property if every closed and bounded subset is compact. Any subset of a Euclidean space, including ...
4
votes
3answers
225 views

Arcwise connected part of $\mathbb R^2$

Helo ! Here's a question that I share: Show that if $D$ is a countable subset of $\mathbb R^2$ (provided with its usual toplogie) then $X=\mathbb R^2 \backslash D $ is arcwise connected
3
votes
2answers
642 views

About connected Lie Groups

How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this ...
4
votes
2answers
714 views

Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?

I have a non homework related question from a text and require a nice clear proof/disproof please Is it true that a subset that is closed in a closed subspace of a topological space is closed in the ...
4
votes
2answers
761 views

Introductory book on Topology [duplicate]

Possible Duplicate: choosing a topology text best book for topology? What book would you recommend for an undergraduate wanting to learn the basics of topology? I've come across several ...
4
votes
3answers
763 views

Example where closure of $A+B$ is different from sum of closures of $A$ and $B$

I need a counter example. I need two subsets $A, B$ of $\mathbb{R}^n$ so that $\text{Cl}(A+ B)$ is different of $\text{Cl}(A) + \text{Cl}(B)$, where $\text{Cl}(A)$ is the closure of $A$, and $A + B = ...
2
votes
2answers
184 views

prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:

$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$ $Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$ $p=\int_{-1}^{1}Z(x)dx$ ...
25
votes
5answers
2k views

Soft Question - Intuition of the meaning of homology groups

I'm studying homology groups and I'm looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible ...
28
votes
6answers
2k views

How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?

I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its ...
11
votes
5answers
2k views

A compact Hausdorff space that is not metrizable

Is there an example of a compact Hausdorff space that is not metrizable? I was thinking maybe the space of continuous functions $f: X \rightarrow Y$ between topological spaces $X, Y$, might work, but ...
10
votes
3answers
794 views

Connectivity, Path Connectivity and Differentiability

I have two questions which pertain to differentiability, connectivity and path connectivity. Ocasionally, I will encounter an author who defines connectivity in the following way: An open subset $U$ ...
17
votes
3answers
2k views

Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$

The following is not a homework problem. I am doing it for self study. Prove that any continuous function from $\mathbb{R}$ to $\mathbb{Q}$ is constant. Here is my proof: Let ...
13
votes
3answers
480 views

is a net stronger than a transfinite sequence for characterizing topology?

For metric spaces, knowledge of the convergence of sequences determines the topology completely. A set is closed in the metric topology if and only if it is closed under the limit of convergent ...
7
votes
2answers
596 views

On the product of separable spaces.

I already figured out how to show that the countable product of separable topological spaces is separable, but I'm out of ideas when the index set has cardinality of $c$. My textbook says it is ...
4
votes
1answer
195 views

Is there exist a homemoorphism between either pair of $(0,1),(0,1],[0,1]$

As the topic is there exist a homeomorphism between either pair of $(0, 1),(0,1],[0,1]$
2
votes
1answer
134 views

prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
7
votes
7answers
1k views

How can I prove formally that the projective plane is a Hausdorff space?

I want to prove the Hausdorff property of the projective space with this definition: the sphere $S^n$ with the antipodal points identified. It's seems easy, but I can't prove formally with this ...
5
votes
3answers
1k views

Every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments

Let $E$ be an open set in $\mathbb{R}$. Fix $x\in E$. I have proved that statement is true when $\{y\in \mathbb{R}|(x,y)\subset E\}$ is bounded above and $\{z\in \mathbb{R}|(z,x)\subset E\}$ is ...
8
votes
5answers
674 views

Can anybody recommend me a topology textbook? [duplicate]

Possible Duplicate: choosing a topology text Introductory book on Topology I'm a graduate student in Math. But I never learnt Topology during my undergraduate study. Next semester, I am ...
4
votes
3answers
152 views

The set of points where two maps agree is closed?

Let $f,g\colon X \to Y$ be continuous maps. Let $Y$ be Hausdorff. Is the set $$A := \{x\in X \, : \, f(x)=g(x) \}$$ necessarily closed ?