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
942 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..
11
votes
5answers
947 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).
13
votes
2answers
542 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} ...
7
votes
1answer
557 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 : ...
36
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.
11
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.
16
votes
1answer
994 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 $F$ ...
4
votes
4answers
932 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
277 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
$$
...
49
votes
5answers
1k 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
2answers
548 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 ...
22
votes
5answers
3k 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
338 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
2k 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
443 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 = ...
1
vote
2answers
169 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$
...
1
vote
2answers
346 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 ...
50
votes
16answers
2k 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 ...
5
votes
2answers
2k 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 ...
2
votes
1answer
69 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 ...
6
votes
4answers
884 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 ...
5
votes
4answers
407 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 ...
2
votes
4answers
585 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 $X \times X$.
...
2
votes
2answers
509 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 ...
17
votes
5answers
1k 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 ...
26
votes
6answers
1k 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 ...
14
votes
8answers
1k views
How to understand compactness? [duplicate]
How to understand the compactness in topology space in intuitive way?
14
votes
5answers
1k views
Why do we require a topological space to be closed under finite intersection?
In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?
I'm assuming ...
7
votes
2answers
2k 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. ...
15
votes
1answer
352 views
Decomposition of a manifold
As a kind of aside to this question, where one of the answers assumed that if $S^n=X \times Y$ then we can assume that $X$ and $Y$ are manifolds.
If we have a manifold $M$, such that $M$ is ...
11
votes
1answer
518 views
(Why) is topology nonfirstorderizable?
Is it the right point of view to say, that topology is nonfirstorderizable (only) because the union of arbitrarily many open sets has to be open? And if "arbitrarily many" was relaxed to "finitely ...
9
votes
5answers
877 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 ...
8
votes
3answers
493 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 $U$ ...
11
votes
3answers
805 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
1answer
475 views
Quotient Space $\mathbb{R} / \mathbb{Q}$
I've just learned about topological quotient spaces and was wondering if anyone can help me with this example I thought of.
Let $(\mathbb{Q}, +)$ be the usual group of rational numbers for addition, ...
5
votes
1answer
803 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 ...
11
votes
3answers
306 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 ...
10
votes
2answers
746 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 $M$ ...
10
votes
2answers
173 views
What operations is a metric closed under?
Suppose $X$ is a set with a metric $d: X \times X \rightarrow \mathbb{R}$. What "operations" on $d$ will yield a metric in return?
By this I mean a wide variety of things. For example, what functions ...
5
votes
2answers
491 views
Where do bitopological spaces naturally occur? Do they have applications?
I am interested where bitopological spaces occur in various parts of mathematics (i.e., what are natural examples of bitopological spaces stemming from various areas of mathematics, not from the ...
7
votes
4answers
344 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 ...
6
votes
7answers
504 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 ...
4
votes
3answers
675 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
435 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
623 views
Notions of equivalent metrics
Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and ...
3
votes
3answers
274 views
preservation of completeness under homeomorphism
Does homeomorphic metric spaces preserves completeness?I mean two metric space which are homeomorphic and one of them is complete$\Rightarrow$ another one is also complete?
0
votes
0answers
384 views
A question about isometry
In Schikoff's "Ultrametric calculus: An introduction to $p$-adic analysis" page 228 exercise 75.A :(On surjectivity of isometries for locally compact $K$)
(i) Show that each isometry of a compact ...
34
votes
5answers
1k views
Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
12
votes
1answer
238 views
Partitioning $\mathbb{R}^2$ into disjoint path-connected dense subsets
Does there exist a partition of the plane into $n=3$ (or more generally $n\ge 3$) disjoint path-connected dense subsets?
Note that the answer is yes if "path-connected" is replaced by "connected", as ...
