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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33
votes
2answers
3k 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 ...
37
votes
16answers
28k 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..
40
votes
11answers
14k views

Any open subset of $\Bbb R$ is a at most 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 ...
11
votes
4answers
4k 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 $X \times X$. ...
15
votes
2answers
2k 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 ...
15
votes
5answers
6k 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 ...
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
1answer
958 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 $$ ...
7
votes
3answers
2k views

For every irrational $\alpha$, the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$

I am not able to prove that this set is dense in $\mathbb{R}$. Will be pleased if you help in a easiest way, $\{a+b\alpha: a,b\in \mathbb{Z}\}$ where $\alpha\in\mathbb{Q}^c$ is a fixed irrational.
11
votes
5answers
2k 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
3k 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 ...
16
votes
7answers
8k 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 ...
64
votes
5answers
6k 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.
6
votes
3answers
573 views

Arcwise connected part of $\mathbb R^2$

Here's a question that I share: Show that if $D$ is a countable subset of $\mathbb R^2$ (provided with its usual topology) then $X=\mathbb R^2 \backslash D $ is arcwise connected.
90
votes
20answers
10k 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 ...
61
votes
4answers
6k views

Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$

It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant. Along similar ...
14
votes
7answers
4k 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.
27
votes
1answer
2k 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 $F$ ...
51
votes
6answers
6k views

Is $[0,1]$ a countable disjoint union of closed sets?

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
36
votes
7answers
4k 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 ...
32
votes
5answers
7k 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 ...
20
votes
6answers
6k 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.
14
votes
5answers
3k 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
1k 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 ...
10
votes
1answer
2k 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
1answer
943 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 : ...
18
votes
1answer
10k views

Continuous function on a compact metric space is uniformly continuous

I am struggling with this question: Prove or give a counterexample: If $f$ is a continuous function on a compact subset $Y$ of a metric space $X$, then $f$ is uniformly continuous on $Y$. ...
29
votes
11answers
2k views

What should be the intuition when working with compactness?

I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in ...
16
votes
2answers
6k 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. ...
19
votes
7answers
4k views

Continuity and Closure

As a part of self study, I am trying to prove the following statement: Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if ...
17
votes
2answers
1k 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 ...
5
votes
4answers
1k views

Nowhere monotonic continuous function

Does there exist a nowhere monotonic continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$? Some nowhere differentiable function sort of object?
43
votes
5answers
5k 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 ...
18
votes
3answers
2k views

Are continuous self-bijections of connected spaces homeomorphisms?

I hope this doesn't turn out to be a silly question. There are lots of nice examples of continuous bijections $X\to Y$ between topological spaces that are not homeomorphisms. But in the examples I ...
21
votes
1answer
2k views

Theorem of Arzelà-Ascoli

The more general version of this theorem in Munkres' 'Topology' (p. 290 - 2nd edition) states that Given a locally compact Hausdorff space $X$ and a metric space $(Y,d)$; a family $\mathcal F$ of ...
17
votes
3answers
3k 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
1answer
2k views

open maps which are not continuous

What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of ...
7
votes
1answer
853 views

Is $\partial (A \cup B)=\partial A\cup\partial B$?

As the topic, assume $\bar A\cap\bar B=\emptyset$. Is $\partial (A \cup B)=\partial A\cup\partial B$, where $\partial A$ and $\bar A$ mean the boundary set and closure of set $A$. I can prove that ...
14
votes
6answers
3k 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
2k 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 ...
1
vote
2answers
577 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 ...
5
votes
2answers
1k 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 ...
18
votes
2answers
1k views

Category-theoretic limit related to topological limit?

Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a ...
5
votes
4answers
4k views

Topologist's sine curve is connected

I just came across the example of the topologist's sine curve that is connected but not path-connected. The rigorous proof of the path-connectedness can be found here.But how can I prove that the ...
13
votes
2answers
2k views

if every continuous function attains its maximum then the (metric) space is compact

Suppose $(M,d)$ a metric space. I want to show that if every continuous real-valued function on $M$ attains a maximum, then the space must be compact. I was trying to do this by assuming $M$ ...
4
votes
6answers
642 views

What's application of Bernstein Set?

Bernstein Set: A subset of the real line that meets every uncountable closed subset of the real line but that contains none of them. It's from wiki. My question is this: How to construct a ...
16
votes
3answers
678 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 ...
12
votes
3answers
3k 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.
11
votes
2answers
1k views

compactness / sequentially compact

I'm looking for two examples: A space which is compact but not sequentially compact A space which is sequentially compact but not compact Explanations why the spaces are compact / not compact and ...
8
votes
4answers
970 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 ...