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Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
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2answers
1k views

Rugged manifold

It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary. Is it still true for topological manifold? Comments: To proof the smooth ...
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2answers
566 views

Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
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2answers
365 views

$\kappa$-homogeneous topological spaces

Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-homogeneous if $|X| \geq \kappa$, and whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$...
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1answer
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Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?

Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$? Remark 1. The answer to the ...
18
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2answers
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Is every compact topological ring a profinite ring?

There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
18
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1answer
655 views

Can two-point sets be Borel?

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...
18
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1answer
625 views

convexity of images of space-filling curves

Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \...
18
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2answers
652 views

Simple connectedness under a metric undistortion condition: on a tricky point in an argument of Gromov

The context I have been reading Gromov's Metric Structures..., and came upon result 1.14.(a), page 11, which states the following. Let $K\subset\mathbb R^d$ be a compact subset, and $d_\ell$ its ...
18
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1answer
663 views

Closed totally disconnected subspaces

It is a remarkable property of uncountable compact metric spaces that each of them contains a homeomorphic copy of the Cantor set. In general, one cannot expect containment of Cantor cubes (in ...
18
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1answer
861 views

Topological transversality

Warmup question: Let us say that two continuous functions $f,g:[0,1]\to \mathbb R$ are topologically transverse if their difference $f-g$ has only finitely many zeros, and each zero separates an ...
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0answers
578 views

The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
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3answers
2k views

How many tacks fit in the plane?

Call a tack the one point union of three open intervals. Can you fit an uncountable number of them on the plane? Or is only a countable number?
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8answers
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Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?

What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found. A good example is Lovász's proof of ...
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10answers
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Undergraduate Topology

I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...
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6answers
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Topological characterization of the closed interval $[0,1]$

This question is related to question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?" but is not exactly equivalent in my opinion. It is even suggested in one ...
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3answers
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“Anti” fixed point property

Let $(X,\tau)$ be a topological space. If $f:X\to X$ is continuous, we say $x\in X$ is a fixed point if $f(x) = x$. The space $(X,\tau)$ is said to have the anti fixed point property (AFPP) if the ...
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2answers
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Example of a compact homogeneous metric space which is not a manifold

A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd ...
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5answers
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Defining a topology in the Power Set

I have the following question: Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$. If the ...
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4answers
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What is the “right” universal property of the completion of a metric space?

I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes: Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...
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5answers
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The “right” topological spaces

The following quote is found in the (~1969) book of Saunders MacLane, "Categories for the working mathematician" "All told, this suggests that in Top we have been studying the wrong mathematical ...
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2answers
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The letters of the word “ART”

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is $...
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1answer
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Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere ?

Is the closed unit ball of the Hilbert space (or, for that matter, of the Hilbert cube, in some metric) homeomorphic to the unit sphere (viz., its own boundary) ? This is clearly uncharacteristic of ...
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2answers
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Is Fürstenberg's topology useful?

It's hard not to be amused and perhaps even amazed when first encountering Fürstenberg's clever "topological" proof that there are infinitely many primes. Closer inspection, however, reveals the ...
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1answer
478 views

Simply connected slices

Assume $\Omega$ is an open set in $\mathbb R^3$ such that the intersection of $\Omega$ with any horizontal plane is simply connected. Can you prove that $\Omega$ is simply connected? (Note that ...
17
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1answer
377 views

Topological embeddings of real projective space in euclidean space

I was wondering whether the real projective space $\Bbb{R}P^n$ embeds topologically into $\Bbb{R}^{n+1}$ for odd $n$. It certainly doesn't for even $n$ because of Alexander duality. Also it doesn't ...
17
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2answers
994 views

compact-open topology on $B(H)$

In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces. Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
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3answers
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How thinly connected can a closed subset of Hilbert space be?

Let H be a separable (and infinite-dimensional) Hilbert space. Is it known whether there exists an infinite subset C of H with the following properties.? (1) C is connected and closed in H. (2) No ...
17
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1answer
358 views

Finite union of closed convex sets is triangulable?

I posted this question on math.stackexchange.com, but didn't get an answer. Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that ...
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1answer
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Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow. In non-Hausdorff topology it is standard to ...
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0answers
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What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
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5answers
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Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?

I'm curious about the following: Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$? Thanks. EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...
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5answers
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Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of Urysohn'...
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5answers
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Which topological spaces are (topological) groups?

General literature does not seem to offer a characterisation of topological groups among all topological spaces. Of course, being completely regular (uniform) is necessary, but separation properties, ...
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3answers
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Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently: In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations $\...
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4answers
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Unique limits of sequences plus what implies Hausdorff?

It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff. What I am ...
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2answers
656 views

Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\...
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1answer
720 views

Can one determine the dimension of a manifold given its 1-skeleton?

This may be an easy question, but I can't think of the answer at hand. Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...
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3answers
873 views

Where else do the (topology) separation axioms turn up?

As an undergraduate I learned point-set topology from Munkres's book, as did many others. One topic that gets a lot of attention is the separation axioms. For example, a space $X$ is normal if any ...
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1answer
510 views

Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?

Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
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1answer
717 views

Is this Wikipedia article linking to the wrong notion of coherent space

I'm reading up on infinite generalizations of the fundamental theorem of distributive lattices. Wikipedia (June 15, 2017) says that there is a duality between distributive lattices and coherent ...
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2answers
770 views

Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural ...
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1answer
349 views

Lowest Dimension for Counterexample in Topological Manifold Factorization

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...
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1answer
527 views

Topological universal algebra: what is a variety?

Very roughly, universal algebra is the study of those classes of algebraic structures which can be defined via a set of equations; such a class is called a variety. Of course there is far more to the ...
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1answer
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Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
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3answers
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Nonseparable example in dimension theory?

Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$? The question closely related to this ...
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1answer
985 views

What if homotopy were expanded to allow any connected space instead of $[0,1]$?

What would happen to homotopy theory if we used a more general definition of homotopy, based on general connected spaces rather than $[0,1]$? Given continuous $f,g:X\to Y$, define $f$ and $g$ to be C-...
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1answer
487 views

A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...