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; ...
20
votes
3answers
893 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 ...
12
votes
13answers
6k 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..
10
votes
5answers
922 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).
34
votes
4answers
3k 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.
12
votes
7answers
2k views
Choosing a topology text
Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
7
votes
1answer
550 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 : ...
16
votes
1answer
966 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 $F$ ...
13
votes
2answers
525 views
No continuous function switches $\mathbb{Q}$ and the irrationals
Is there a way to prove the following result using connectedness?
Result:
Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} ...
4
votes
4answers
908 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 ...
2
votes
1answer
263 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
$$
...
14
votes
2answers
525 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 ...
21
votes
5answers
2k 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 ...
2
votes
2answers
324 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 ...
7
votes
6answers
1k 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.
4
votes
3answers
425 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 = ...
