Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [continuum-theory]

For questions from continuum theory. A continuum is a compact connected metric space (sometimes this term is used for a compact connected Hausdorff space). Do not use this tag for questions related to the Continuum Hypothesis in Set Theory.

Continuum Theory is the study of nonempty, connected, compact, metric spaces, referred to as Continua. In some more general contexts, a continuum is any nonempty, connected, compact, Hausdorff space.

Herein lies the study of spaces such as the Sierpinski Carpet, the Warsaw Circle, and the Topologists' Sine Curve as well as various fixed point theorems.

See also:

Do not use this tag for questions related to the Continuum Hypothesis in Set Theory.

98 questions
3
votes
1 answer

Brouwer Reduction Theorem

I went looking for a statement of the Brouwer Reduction Theorem, but Google only gives hits for his fixed point theorem. I talked to an old professor of mine about it being used to prove if you have two mutually exclusive closed sets, there exists…
Tom Kacvinsky
  • 31
1
vote
1 answer

Naming a property of continua

Let me define a continuum $X$ to be nice if for each pair $a,b\in X$ of distinct points, there are subcontinua $M,N\subseteq X$ such that: $a\in M\setminus N$; $b\in N\setminus M$; and $M\cup N=X$. (This is a very strong form of aposyndesis.) If a…
jinpa
  • 38
  • 7
0
votes
0 answers

The number of components of preimage of a continuum under a polynomial

Given a ploynomial $f$ with degree $d$, then we have a dynamical system $(\mathbb{C},f)$, the critical points are those points $z$ such that the derivative of $f$ at which is zero. If we have a proper continuum $N\subset\mathbb{C}$, then we know…
Yee Neil
  • 91