📂Topological Data Analysis

Continuity of Relative Homotopy 연속함수의 상대적 호모토피

Definition ¹

Generalization of Homotopy

1. Let $I = [0,1]$ be the unit interval and $X, Y$ a topological space. For two continuous mappings $f_{0} , f_{1} : X \to Y$, if there exists a continuous mapping $F : X \times Y$ satisfying $$ F (x , 0) = f_{0} (x) \\ F (x , 1) = f_{1} (x) $$ then $f_{0}, f_{1}$ is said to be homotopic and $F$ is called a homotopy between $f_{0}$ and $f_{1}$.

Relative Homotopy

2. For a subset $A \subset X$ of $X$, $$ F(a,t) = f_{0} (a) \qquad , \forall a \in A , \forall t \in I $$ if there exists a homotopy $F : X \times I \to Y$ between $f_{0}$ and $f_{1}$ satisfying this, then $f_{0}$ and $f_{1}$ are said to be relatively homotopic to $A$.

Explanation

1. The generalization of homotopy is simply the generalization of homotopy defined on paths to continuous mappings. Just as a homotopy like $$ F : I \times I \to Y $$ existed between two paths $f_{0} : I \to Y$ and $f_{1} : I \to Y$, now the domain earlier represented by the interval $I = [0,1]$ has merely been extended to a general topological space $X$ as follows. $$ F : X \times I \to Y $$
2. According to the definition of relative homotopy, at all $a \in A$ it is $f_{0} (a) = f_{1} (a)$, and $F$ is called a relative homotopy to $A$ and can be expressed as $f_{0} \simeq_{\text{rel } A} f_{1}$ or $f_{0} \simeq\ f_{1} (\text{rel } A)$. The idea that a homotopy is relative simply means that at $A$, it remains unchanged. Naturally, to remove the term ‘relative’ from the definition, it would suffice to have just $A = \empty$.
3. In actual literature, the most common use of relative homotopy refers to homotopy itself. One can often see notation like the following to indicate that it only equals at the endpoints $\left\{ 0,1 \right\}$ of the unit interval $[0,1]$. $$ f \simeq_{\left\{ 0,1 \right\}} g $$