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Results tagged with gn.general-topology
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user 2362
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
13
votes
0
answers
255
views
Big list of Hochster dual concepts
Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and …
7
votes
2
answers
197
views
Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold interse...
Cover the $n$-disc with $n+1$ open sets $D^n = U_0 \cup \dotsb \cup U_n$. Suppose that $U_0 \cap \dotsb \cap U_n = \emptyset$. Suppose moreover that the cover is irredundant in the sense that no prope …
11
votes
1
answer
709
views
On the classification of second-countable Stone spaces
Let $X$ be a Stone space (i.e. totally disconnected compact Hausdorff). Then the following are equivalent:
$X$ is second countable
$X$ is metrizable
$X$ has countably many clopen subsets
$X$ is an in …
5
votes
0
answers
131
views
Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with …
6
votes
1
answer
451
views
Which maps of topological spaces have the right lifting property with respect to all split m...
Let $p : X \to Y$ be a continuous map. We say that $p$ has the right lifting property with respect to split monomorphisms if, for every space $B$, and every retract $A \subseteq B$, and for every cont …
1
vote
0
answers
99
views
"Classifying" causally closed sets in Minkowski space
Let $M = \mathbb R^{D+1}$ be Minkowski space. Recall that the causal complement of a set $A \subseteq M$ is the set $A^\perp \subseteq M$ where $p \in A^\perp$ there is no timelike path between $p$ an …
6
votes
0
answers
187
views
What is a non-smooth connection?
Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $ …
5
votes
0
answers
203
views
What are all of the topological (commutative) monoid structures on a closed interval?
Consider a closed real interval $[a,b]$ as a toplogical space. Up to homeomeorphism it doesn't matter, but I like to take $[a,b] = [0,\infty]$.
Question 1: What are all of the topological commutative …
4
votes
1
answer
796
views
How do finite door spaces work?
Recall that a door space is a topological space where every set is either open or closed (or both). A topological space is finite if it has finitely many points. I'm interested in learning about finit …
15
votes
1
answer
467
views
What are the algebras for the ultrafilter monad on topological spaces?
Motivation: Let $(X,\tau)$ be a topological space. Then the set $\beta X$ of ultrafilters on $X$ admits a natural topology (cf. Example 5.14 in Adámek and Sousa - D-ultrafilters and their monads), giv …
7
votes
0
answers
262
views
When is the exponential of a map proper?
Let $X$ be a compact Hausdorff space. Then if $f: A \to B$ is a map between discrete spaces, the induced map $f^\ast: X^B \to X^A$ is proper.
Question: Are there other classes of map $f: A \to B$ suc …
36
votes
1
answer
3k
views
Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of …
15
votes
3
answers
1k
views
What do absolute neighborhood retracts look like?
In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. Thi …
1
vote
1
answer
437
views
Topology of length spaces
How wild can the topology of a length space be? That is,
Questions:
Let $X$ be a metric space where the distance between two points $x,y \in X$ is the infinum of lengths of rectifiable paths from …
40
votes
3
answers
3k
views
A map of non-pathological topology?
I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the low …