# Questions tagged [gn.general-topology]

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

2,951 questions
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### Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology book, James Munkres makes an interesting remark: It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
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### Cryptomorphisms

I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are Topological Spaces. These can be defined in terms of open sets, ...
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### complement of a totally disconnected closed set in the plane

While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
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### Countable connected Hausdorff space

Let me start by reminding two constructions of topological spaces with such exotic combination of properties: 1) The elements are non-zero integers; base of topology are (infinite) arithmetic ...
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### On the global structure of the Gromov-Hausdorff metric space

This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at math....
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### The Closure-Complement-Intersection Problem

Background Let $A$ be a subset of a topological space $X$. An old problem asks, by applying various combinations of closure and complement operations, how many distinct subsets of $X$ can you ...
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### An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request

There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter. But there's ...
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The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of $... 0answers 648 views ### Are amenable groups topologizable? I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group$G$is ... 6answers 3k views ### Is there a classification of open subsets of euclidean space up to homeomorphism? I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ... 2answers 1k views ### Colimits in the category of smooth manifolds In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of ... 3answers 2k views ### What is a TMF in topology? What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms? 4answers 2k views ### Is every locally connected subset of Euclidean space R^n locally path connected ? This is not actually a question asked by me. But since I do not know the answer, I would love to know if someone here could answer it. 1answer 2k views ### connectivity of the group of orientation-preserving homeomorphisms of the sphere In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written: Is the group of orientation-preserving ... 5answers 1k views ### Explanation for E_8's torsion To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of$SO(n)$'s is ... 3answers 3k views ### Weak and Strong Integration of vector-valued functions This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference: Suppose$f:X\to E$is a continuous function from a compact spaces (endowed ... 4answers 696 views ### Problems for developing mathematical visualization expertise Einstein stated that he often explored and reasoned visually and spatially, and only after achieving understanding cast his insights into algebraic form. He could just "see" the answer. There are ... 1answer 654 views ### Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related? It is known that there are only 14 reasonable tensor norms in$Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ... 1answer 988 views ### Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras? Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra. Moreover, a morphism of commutative von Neumann algebras induces a continuous morphism of the corresponding ... 0answers 386 views ### Disc bounded by a plane curve Let$\Sigma$be a sphere topologically embedded into$\mathbb{R}^3$. Is it always possible to find a disc$\Delta\subset\Sigma$which is bounded by a plane curve? It is easy to find an open disc ... 15answers 14k views ### Learning Topology EDIT (Harry): Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ... 6answers 8k views ### Is a topology determined by its convergent sequences? Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the ... 13answers 14k views ### Examples of non-metrizable spaces I want to know some examples of topological spaces which are not metrizable. Of course one can construct a lot of such spaces but what I am looking for really is spaces which are important in other ... 4answers 5k views ### Can you explicitly write$\mathbb{R}^2$as a disjoint union of two totally path disconnected sets? An anonymous question from the 20-questions seminar: Can you explicitly write$\mathbb{R}^2$as a disjoint union of two totally path disconnected sets? 3answers 5k views ### sets with positive Lebesgue measure boundary Consider a compact subset$K$of$R^n$which is the closure of its interior. Does its boundary$\partial K$have zero Lebesgue measure ? I guess it's wrong, because the topological assumption is ... 3answers 1k views ### Non-homeomorphic spaces such that taking away a point makes them homeomorphic Are there topological spaces$X,Y$, each having more than$2$points, satisfying the following two properties?$X\not\cong Y$, and there is a bijection$\varphi: X\to Y$such that for all$x\in X$... 4answers 3k views ### Why are topological ideas so important in arithmetic? For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in ... 4answers 2k views ### Compact open topology on$\mathrm{Homeo}(X)$Let$X$and$Y$be topological spaces. Define the compact open topology on the set$\mathrm{M}(X,Y)$of continuous maps from$X$to$Y$via the subbase$[K,O]$of all maps$f:X\rightarrow Y$s.t.$f(K)...
Suppose $f: S^2 \rightarrow {\bf R}^2$ is continuous; let $A$ be the set of points $u \in S^2$ such that $f(u)-f(-u) \in {\bf R} \times \{0\}$ (where $-u$ denotes the antipode of $u$). Given \$u,-u \in ...