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; ...
273
votes
6answers
60k views
Why can you turn clothing right-side-out?
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the ...
64
votes
2answers
6k views
Topology: The Board Game
Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website. Any comments on game play can be incorporated into ...
54
votes
2answers
2k views
Does this property characterize a space as Hausdorff?
As a result of this question, I've been thinking about the following condition on a topological space $Y$:
For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, ...
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 ...
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 $\mathbb{R}^n$. ...
40
votes
15answers
3k views
Why can't you flatten a sphere?
It's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old?
As soon as an explanation starts using terms like "Gaussian ...
40
votes
3answers
715 views
Trying to define $\mathbb{R}^{0.5}$ topologically [duplicate]
A few days ago, I was trying to generalize the defintion of Euclidean spaces by trying to define $\mathbb{R}^{0.5}$.
Question: Is there a metric space $A$ such that $A\times A$ is homeomorphic to ...
39
votes
4answers
1k views
Is the box topology good for anything?
In point-set topology, one always learns about the box topology: the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where $U_i ...
38
votes
7answers
1k views
Why is one “$\infty$” number enough for complex numbers?
Can anyone give me a rigorous explanation, why one needs only one number "$\infty$", when dealing with complex numbers, instead of 2 numbers $+\infty, \ -\infty$ like in the case, when dealing with ...
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.
35
votes
7answers
2k views
What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?
Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) ...
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 ...
33
votes
5answers
3k 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?
32
votes
6answers
1k views
Why is compactness in logic called compactness?
In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set.
Most logic texts either don't explain the terminology, or ...
30
votes
3answers
2k 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 ...
28
votes
3answers
2k views
Why is the Möbius strip not orientable?
I am trying to understand the notion of an orientable manifold.
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
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 ...
26
votes
1answer
564 views
A Universal Property Defining Connected Sums
I once read (I believe in Ravi Vakil's notes on Algebraic Geometry) that the connected sum of a pair of surfaces can be defined in terms of a universal property. This gives a slick proof that the ...
23
votes
3answers
807 views
If every continuous $f:X\to X$ has $\text{Fix}(f)\subseteq X$ closed, must $X$ be Hausdorff?
Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$.
In a recent comment, I wondered whether
$X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is closed for every continuous ...
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 ...
22
votes
5answers
1k views
Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”
In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It ...
22
votes
3answers
491 views
Why is the Hilbert Cube homogeneous?
The Hilbert Cube $H$ is defined to be $[0,1]^{\mathbb{N}}$, i.e., a countable product of unit intervals, topologized with the product topology.
Now, I've read that the Hilbert Cube is homogeneous. ...
22
votes
3answers
415 views
Is $\mathbb{R}^{\infty}$ homeomorphic to $\mathbb{R}^{\infty}\setminus\{0\}$?
Let $\mathbb{R}^{\infty}$ be a linear topological space of all sequences
$x=(x_{1},x_{2},\ldots,x_{n},\ldots)$ of real numbers with a product topology,
or, in other words, let $\mathbb{R}^{\infty}$ be ...
22
votes
1answer
218 views
If $S \times \Bbb{R}$ is homeomorphic to $T \times \Bbb{R}$ and $S$ is compact, can we conclude that $T$ is compact?
Suppose $S$ and $T$ are connected manifolds such that:
1) $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$,
2) $S$ is compact.
can we conclude that $T$ is compact?
22
votes
1answer
539 views
Showing a filter on the Power set of $\mathbb{Z}$ is a one point Filter
Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
21
votes
10answers
5k views
Real life applications of Topology
The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever.
I want to disprove him, so posting the question here
What ...
21
votes
1answer
572 views
What is the cardinality of the set of all topologies on $\mathbb{R}$?
This was asked on Quora. I thought about it a little bit but didn't make much progress beyond some obvious upper and lower bounds. The answer probably depends on AC and perhaps also GCH or other ...
21
votes
4answers
667 views
Correspondences between Borel algebras and topological spaces
Though tangentially related to another post on MathOverflow (here), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...
Suppose ...
21
votes
1answer
253 views
If $S \times \Bbb{R}^k$ is homeomorphic to $T \times \Bbb{R}^k$ and $S$ is compact, can we conclude that $T$ is compact?
If $S \times \Bbb{R}^k$ is homeomorphic to $T \times \Bbb{R}^k$ for some $k \geq 1$ and $S$ is compact, can we conclude that $T$ is compact?
The case where $k=1$ and $S$ and hence also $T$ are ...
21
votes
2answers
355 views
Existence of non-constant continuous functions
Under what circumstances is there at least one non-constant continuous function from a topological space $X$ to a topological space $Y$? Assume that $X$ and $Y$ each have at least two points. If $X$ ...
20
votes
3answers
708 views
Are the rationals minus a point homeomorphic to the rationals?
A while ago I was dreaming up point-set topology exam questions, and this one came to mind:
Is $\mathbb Q\setminus \{0\}$ homeomorphic to $\mathbb Q$? (Where both sets have the subspace topology ...
20
votes
3answers
946 views
Set of continuity points of a real function
I have a question about subsets $$
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 ...
20
votes
1answer
217 views
When is Stone-Čech compactification the same as one-point compactification?
For the space $\omega_1$ (with the order topology) we have $\beta\omega_1=\omega_1+1$ (or $\beta[0,\omega_1)=[0,\omega_1]$, if you prefer this notation), i.e., it is an example of a space for which
...
20
votes
1answer
884 views
Is this space contractible?
Let $X$ be the following topological space (with the subspace topology): Connect the rational points of $([0,1]\cap \mathbb{Q})\times \{0\}$ with the point $(0,1)$ and connect the points of ...
19
votes
4answers
916 views
The graph of xy = 1 is connected or not
The graph of $xy = 1$ in $\Bbb C^2$ is connected. True or false?
I know that it is not connected in $\Bbb R^2$, but what is the case of $\Bbb C^2$?
19
votes
4answers
357 views
Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected
Math people:
In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
19
votes
2answers
1k views
Is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$
I need a hint. The problem is: is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$
I'm pretty sure that there aren't any, but so far I couldn't find the proof.
My best idea so far is ...
19
votes
1answer
493 views
Homeomorphism preserving distance
I have a problem but I don't know if there is a solution or a counter-example.
Problem: Let $M$ be a non trivial compact connected metric space and let $f:M\to M$ be a homeomorphism. Show that there ...
19
votes
1answer
383 views
An interesting topological space with $4$ elements
There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
18
votes
3answers
471 views
Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space
Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- ...
18
votes
2answers
318 views
What's the difference between rationals and irrationals - topologically?
I know that sets of rational and irrational numbers are quite different. In measure, almost no real number is rational and of course, $\mathrm{card}(\mathbb Q) < \mathrm{card}(\mathbb R \setminus ...
18
votes
4answers
258 views
Continuous $f\colon [0,1]\to \mathbb{R}$ all of whose nonempty fibers are countably infinite?
I have been told it is possible to construct a continuous function $f\colon [0,1]\to \mathbb{R}$ such that $f^{-1}(x)$ is either empty or has cardinality $\aleph_0$ for every $x\in \mathbb{R}$. I've ...
18
votes
3answers
876 views
Size of the closure of a set
Why in a Hausdorff sequentially compact space the size of the closure of a countable subset is less or equal than $c$ ? I can see why this is true when the space if first countable but we are not ...
18
votes
2answers
317 views
What is a metric for $\mathbb Q$ in the lower limit topology?
A useful source of counterexamples in topology is $\mathbb R_\ell$, the set $\mathbb R$ together with the lower limit topology generated by half-open intervals of the form $[a,b)$. For example this ...
18
votes
1answer
424 views
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
18
votes
0answers
143 views
Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?
This question came up in a recent video series of lectures by Mike Freedman available through Max Planck Institut's website. He proves the "difficult" converse direction, that $X\times \mathbb R\cong ...
17
votes
8answers
2k views
Why do introductory real analysis courses teach bottom up?
A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
17
votes
6answers
820 views
Uncountable closed set of irrational numbers
Could you construct an actual example of a uncountable set of irrational numbers that is closed (in the topological sense)?
I can find countable examples that are closed, like $\{ \sqrt{2} + ...
17
votes
4answers
435 views
Is the closure of a Hausdorff space, Hausdorff?
$(X,\mathcal T)$ is a topological space which has a dense Hausdorff subspace. Is $X$ Hausdorff?
17
votes
2answers
514 views
What is algebraic geometry?
I am a second year physics undergrad, loooking to explore some areas of pure mathematics. A word that often pops up on the internet is algebraic geometry.
What is this algebraic geometry exactly? ...
