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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29
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 ...
31
votes
16answers
20k 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..
18
votes
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).
10
votes
4answers
5k 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 ...
4
votes
1answer
752 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
4answers
3k 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 $X \times X$. ...
27
votes
8answers
9k 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 ...
56
votes
5answers
5k 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.
11
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 ...
6
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.
72
votes
20answers
8k 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 ...
13
votes
7answers
3k 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.
10
votes
5answers
1k 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 ...
15
votes
7answers
6k 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 ...
50
votes
4answers
5k 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 ...
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 ...
25
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 $F$ ...
12
votes
6answers
4k 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.
8
votes
1answer
839 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 : ...
6
votes
3answers
392 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.
17
votes
2answers
896 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 ...
27
votes
5answers
5k 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 ...
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 ...
10
votes
1answer
7k 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 $Y$. ...
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 ...
34
votes
7answers
3k 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 ...
46
votes
5answers
5k 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?
12
votes
2answers
5k 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. ...
12
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 ...
7
votes
3answers
974 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 ...
5
votes
4answers
861 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?
7
votes
2answers
765 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 ...
6
votes
3answers
1k views

Proof of the Lebesgue number lemma

I want to prove the Lebesgue number lemma: Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, there exists $\delta \gt 0$ such that for each subset of $X$ ...
8
votes
4answers
661 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
223 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 ?
1
vote
2answers
512 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 ...
60
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 $\mathbb{R}^n$. ...
37
votes
5answers
3k 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 ...
14
votes
8answers
2k views

How to understand compactness? [duplicate]

How to understand the compactness in topology space in intuitive way?
14
votes
7answers
3k 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
884 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 $B$, ...
20
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 ...
12
votes
4answers
3k views

Why does a convex set have the same interior points as its closure?

Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just ...
12
votes
3answers
2k 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.
3
votes
3answers
3k 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 ...
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 ...
14
votes
3answers
591 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 ...
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 ...
4
votes
1answer
500 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 ...
9
votes
2answers
858 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 ...