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Topology the study of objects that can be continuously deformed into other objects without tearing or making holes in the object. It can also mean a family of sets that have the property of a topological space. The properties are convergence, connectedness, and continuity.

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Do Banach spaces and linear contraction maps form a model of ILL with an exponential?

Recently, I read on the nLab that the category of Banach spaces and linear contractions is small complete, small cocomplete, and monoidal closed. This means that Banach spaces and short linear maps ...
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1answer
92 views

Topology/Space of Recursive Algebraic Datatypes

I have a recursive algebraic datatype. I (somewhat arbitrarily) defined one function to compute distance between instances, and am trying to define a function to approximate a "vector" between ...
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1answer
129 views

How to design an algorithm which turns an undirected graph into directed with all nodes of indegree higher than 0? [closed]

Given an undirected graph $G=(V,E)$ devise an algorithm that will check whether its edges can be directed in such a way that the vertices of the resulting directed graph will all have indegree higher ...
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0answers
94 views

Structures obtained by gluing simplices

I'm looking for the correct name of geometric structures obtained as follows. 2-structures: A collection $X$ of triangles is a $2$-structure. If $X$ is a $2$-structure and $Y$ is obtained from $X$ ...
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0answers
183 views

Applications of “Seemingly Impossible Functional Programs”

What are some practical applications (existing or potential) for Martin Escardo's "Seemingly Impossible Functional Programs"? For starters, here are a few from: Alex Simpson’s Lazy functional ...
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0answers
90 views

Approximating liveness properties with safety properties

Given a finite alphabet set $\Sigma$, the set $\Sigma^{\omega}$ of infinite words over $\Sigma$ can be topologized with a metric $d: \Sigma^{\omega} \rightarrow \mathbb{R}$ such as: $\forall w_1, w_2 ...
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2answers
210 views

The complexity of finding a Borsuk-Ulam point

The Borsuk-Ulam theorem says that for every continuous odd function $g$ from an n-sphere into Euclidean n-space, there is a point $x_0$ such that $g(x_0)=0$. Simmons and Su (2002) describe a method ...
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3answers
416 views

Applications for set theory, ordinal theory, infinite combinatorics and general topology in computer science?

I am a mathematician interested in set theory, ordinal theory, infinite combinatorics and general topology. Are there any applications for these subjects in computer science? I have looked a bit, and ...
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2answers
1k views

Why is “topological sorting” topological?

Why is "topological sorting" called "topological"? Is it just because it determines an order without altering any vertices or edges -- like a doughnut and coffee cup are topologically equivalent? Why ...
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0answers
117 views

Example of non-disk bounding planarly nested sequences of cycles

I am trying to find an example for the Theorem 5.1 of the paper "Combinatorial Local Planarity and the Width of Graph Embeddings" that can be found at http://www.fmf.uni-lj.si/~mohar/Reprints/1992/...
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1answer
705 views

To what extent can the mathematics of Reals be applied to Computable Reals?

Is there a general theorem that would state, with proper sanitization, that most known results regarding the use of real numbers can actually be used when considering only computable reals? Or is ...
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2answers
466 views

Would a purely topological computational model be useful in decision problems in topology?

If one were to develop a purely topological computational model based upon the equivalence of information in knots and the model would perform transformations of that information. This would be the ...
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3answers
564 views

Barcode of a graph

Using persistent homology, we can analyze the (topological) shape of a cloud of points using the following three-step method: convert the point set into a simplicial complex (and there are a few ...
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1answer
134 views

Generalizations of planar graphs that include hypercubes with large side length in $R^d$

A lot of people have asked about generalizations of planar graphs on other forums. Some topics include: https://mathoverflow.net/questions/7650/generalizations-of-planar-graphs https://math....
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1answer
239 views

How do you compute the fixed point of a best-response function efficiently?

I have a polynomial time best-response function that has the same properties as a game-theory game (convexity, compactness, set-valued). I don't know that much topology, but my understanding is that ...
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1answer
319 views

Adherence of languages and the Dyck language

Let $L \subseteq X^*$ and $X = \{a,b\}$ be a language of finite words, denote by $A(u)$ the prefixes of some word (finite or infinite), then the adherence $\mbox{Adh}(L)$ is defined to be the set of ...
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0answers
107 views

Transfering properties from subsets of $X^*$ to subsets of $X^{\omega}$ by using the topology induces by Cantor space

A language $L \subseteq X^*$ is non-counting of order $n > 0$ iff for all $u,v, w \in X^*$ $$ uv^nw \in L \Leftrightarrow uv^{n+1} w \in L. $$ A $\omega$-language (set of infinite sequences) $L \...
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1answer
251 views

Is there a geometrical picture for adiabatic quantum computation?

In adiabatic quantum computation (AQC), one encodes the solution to an optimization problem in the ground state of a [problem] Hamiltonian $H_p$. To get to this ground state, you start in an easily ...
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0answers
97 views

test if a polygon-surface is closed (+ additional nice-to-have requirements)

I have a 3D object bounded by Polygons. Is there a standard algorithm that tests if the volume is closed e.g. no polygon is missing? Example: I have a cube bounded by six squares. The algorithm ...
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2answers
192 views

Relation between vertices, cells, and vertex-cell-incidences in 3D subdivisions

Consider a planar subdivision, with F faces, V vertices, E edges, and I face-vertex incidences. For simplicity, assume a "non-degenerate" situation in which each vertex occurs on the boundary cycle of ...
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1answer
273 views

Is there a characteristic function of a tree?

Consider a set of trees $T=\{T_{\alpha}\}$, and for any $T_{\alpha}\in T$, $T_{\alpha}$ has $n$ nodes. Can we find a ‘characteristic’ function $f:T\longmapsto{\mathbb{R}}$ describing trees' ...
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1answer
618 views

A topological space related to SAT: is it compact?

The Satisfiability problem is, of course, a fundamental problem in theoretical CS. I was playing with one version of the problem with infinitely many variables. $\newcommand{\sat}{\mathrm{sat}} \...
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1answer
356 views

Place n points in a box as far away from each other as possible

Can you suggest an optimal or heuristic algorithm for placing points on a 2D plane (within a constrained space) such that minimum distance between any two points is maximized. In other words, I'm ...
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1answer
204 views

Computational Library to compute Quantum Cluster States

I want to write a simulator for a quantum computing model that I am working on and I was wondering what would be the correct library / implementation strategy to implement quantum cluster states? ...
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2answers
1k views

Complexity of Topological Properties.

I am a computer scientist taking a course on Topology (a sprinkling of point-set topology heavily flavored with continuum theory). I have become interested in decision problems testing a description ...
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1answer
353 views

In domain theory, what can the extra structure present in metric spaces be used for?

Smyth's chapter in the handbook of logic in computer science and other references describe how metric spaces can be used as domains. I do understand that complete metric spaces give unique fixed ...
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14answers
10k views

Applications of topology to computer science

I'd like to write a survey on the applications of Topology in Computer Science. I plan to cover the history of topological ideas in Computer Science and also highlight a few current developments. It ...
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3answers
727 views

Is the 3-sphere recognition problem NP-complete?

It is known that determining whether or not a given triangulated 3-manifold is a 3-sphere is in NP, via work by Saul Schleimer in 2004: "Sphere recognition lies in NP" arXiv:math/0407047v1 [math.GT]. ...
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2answers
625 views

Bounded-input bijections of infinite sequences

Here is a puzzle I haven't managed to solve. I would like to know if this problem is already known, or has an easy solution. It is possible to define a bijection $ 3^\mathbb{N} \cong 5^\mathbb{N} $ ...
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20answers
7k views

Examples of “Unrelated” Mathematics Playing a Fundamental Role in TCS?

Please list examples where a theorem from mathematics which was not normally considered to apply in computer science was first used to prove a result in computer science. The best examples are those ...
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3answers
455 views

What applications of Cantor space are there?

Are there well-established applications of the Cantor space ($2^\omega$) in computer science, other than those connected with computable real arithmetic? John Tucker's page Computation on Topological ...