SOME PROPERTIES OF PROXIMAL HOMOTOPY THEORY

Consider a pseudo-metric space $(X,d)$. A binary relation $\delta$ defined by

satisfies

for $D(E,F)=\inf\{d(x_{1},x_{2}):x_{1}\in E,x_{2}\in F\}$[1, 21, 5].

$\delta$ is a nearness relation and $E\delta F$ is read as “$E$ is near $F$”. Otherwise, the notation $E\underline{\delta}F$ means that “$E$ is far from $F$”.

As an example of a proximity space, the discrete proximity $\delta$ on a (nonempty) set $X$ is defined by “$E\delta F\ \Leftrightarrow\ E\cap F\neq\emptyset$” for $E$, $F\subset X$. Also, the indiscrete proximity $\delta^{{}^{\prime}}$ on a (nonempty) set $X$ is given by $E\delta^{{}^{\prime}}F$ for any nonempty subsets $E$ and $F$ in $X$. A subset $E$ of $X$ with a proximity $\delta$ is a closed set if “$x\delta E\ \Rightarrow\ x\in E$”. The converse is also valid. Therefore, given a proximity $\delta$ on $X$, a topology $\tau(\delta)$ is defined by the family of complements of all closed sets via Kuratowski closure operator[5].

Given any proximities $\delta$ and $\delta^{{}^{\prime}}$ on respective sets $X$ and $X^{{}^{\prime}}$, a map $h$ from $X$ to $X^{{}^{\prime}}$ is called proximally continuous if “$E\delta F\ \Rightarrow\ h(E)\delta^{{}^{\prime}}h(F)$” for $E$, $F\subset X$[1, 21]. We denote a proximally continuous map by “pc-map”. Given a proximity $\delta$ on $X$ and a subset $E\subset X$, a subspace proximity $\delta_{E}$ is defined on the subsets of $E$ as follows[5]: “$E_{1}\delta E_{2}\ \Leftrightarrow\ E_{1}\delta_{E}E_{2}$” for $E_{1}$, $E_{2}\subset E$. Let $(X,\delta)$ be a proximity space and $(E,\delta_{E})$ a subspace proximity. A pc-map $k:(X,\delta)\rightarrow(E,\delta_{E})$ is a proximal retraction provided that $k\circ j$ is an identity map on $1_{E}$, where $j:(E,\delta_{E})\rightarrow(X,\delta)$ is an inclusion map.

We say that $h$ is a proximity isomorphism provided that $h$ is a bijection and each of $h$ and $h^{-1}$ is pc-map[5]. According to this, $(X,\delta)$ and $(X^{{}^{\prime}},\delta^{{}^{\prime}})$ are said to be proximally isomorphic spaces. Another important proximity relation is given on the subsets of the cartesian product of two proximity spaces as follows[3]: Let $\delta$ and $\delta^{{}^{\prime}}$ be any proximities on respective sets $X$ and $X^{{}^{\prime}}$. For any subsets $E_{1}\times E_{2}$ and $F_{1}\times F_{2}$ of $X\times X^{{}^{\prime}}$, $E_{1}\times E_{2}$ is near $F_{1}\times F_{2}$ if $E_{1}\delta F_{1}$ and $E_{2}\delta^{{}^{\prime}}F_{2}$.

The map $F$ in Definition 2.4 is said to be a proximal homotopy between $h$ and $h^{{}^{\prime}}$. We simply denote a proximal homotopy by “prox-hom”. Similar to topological spaces, prox-hom is an equivalence relation on proximity spaces. Let $\delta$ be a proximity on $X$ and $E\subset X$. $E$ is called a $\delta-$neighborhood of $F$, denoted by $F\ll_{\delta}E$, provided that $F\underline{\delta}(X-E)$[5]. The proximal continuity of any function $h:(X,\delta)\rightarrow(X^{{}^{\prime}},\delta^{{}^{\prime}})$ can also be expressed as

for any $E$, $F\subset X^{\prime}$.

The proximal continuity of the proximal path $h:I\rightarrow X$ in Definition 2.6 means that “$D(E,F)=0\ \Rightarrow\ h(E)\delta h(F)$” for $E$, $F\in I$. Recall that $X$ is a connected proximity space if and only if for all nonempty $E$, $F\in\mathcal{P}(X)$, $E\cup F=X$ implies that $E\delta F$[7]. Let $\delta$ be a proximity on $X$. Then $X$ is called a path-connected proximity space if, for any points $x_{1}$ and $x_{2}$ in $X$, there exists a proximal path from $x_{1}$ to $x_{2}$ in $X$.

Assume that $X$ is a nonempty set and $x\in X$. Consider the set $\Phi=\{\phi_{1},\cdots,\phi_{m}\}$ of maps (generally named as probe functions) $\phi_{j}:X\rightarrow\mathbb{R}$, $j=1,\cdots,m$, such that $\phi_{j}(x)$ denotes a feature value of $x$. Let $E\subset X$. Then the set of descriptions of a point $e$ in $E$, denoted by $\mathcal{Q}(E)$, is given by the set $\{\Phi(e):e\in E\}$, where $\Phi(e)$ (generally called a feature vector for $e$) equals $(\phi_{1}(e),\cdots,\phi_{m}(e))$. For $E$, $F\subset X$, the binary relation $\delta_{\Phi}$ is defined by

and $E\delta_{\Phi}F$ is read as “$E$ is descriptively near $F$”[9, 10, 12]. Also, $E\underline{\delta_{\Phi}}F$ is often used to state “$E$ is descriptively far from $F$”. The descriptive intersection of $E$ and $F$ and the descriptive union of $E$ and $F$ are defined by

and

respectively[12].

A binary relation $\delta_{\Phi}$ defined by (1) [6] satisfies

$\delta_{\Phi}$ is a descriptive nearness relation.

A map $h:(X,\delta_{\Phi})\rightarrow(X,\delta_{\Phi}^{{}^{\prime}})$ is called descriptive proximally continuous provided that “$E\delta_{\Phi}F\ \Rightarrow\ h(E)\delta_{\Phi}^{{}^{\prime}}h(F)$” for $E$, $F\subset X$[13, 14]. We denote a descriptive proximally continuous map by “dpc-map”. Let $\delta_{\Phi}$ be a descriptive proximity on $X$, and $E\subset X$ a subset. Then a descriptive subspace proximity $\delta_{\Phi}^{E}$ is defined on the subsets of $E$ as follows:

for $E_{1}$, $E_{2}\subset E$. Given a descriptive proximity $\delta_{\Phi}$ on $X$, a descriptive subspace proximity $(E,\delta_{\Phi}^{E})$, and the inclusion $j:(E,\delta_{\Phi}^{E})\rightarrow(X,\delta_{\Phi})$, a dpc-map $k:(X,\delta_{\Phi})\rightarrow(E,\delta_{\Phi}^{E})$ is called a descriptive proximal retraction if $k\circ j=1_{E}$.

$h$ is a descriptive proximity isomorphism if $h$ is a bijection and each of $h$ and $h^{-1}$ is dpc-map[5]. Hence, $(X,\delta_{\Phi})$ and $(X^{{}^{\prime}},\delta_{\Phi}^{{}^{\prime}})$ are called descriptive proximally isomorphic spaces. A descriptive proximity relation on the cartesian product of descriptive proximity spaces is defined as follows[3]: Assume that $\delta_{\Phi}$ and $\delta_{\Phi}^{{}^{\prime}}$ are any descriptive proximities on $X$ and $X^{{}^{\prime}}$, respectively. $E\delta_{\Phi}F$ and $E^{{}^{\prime}}\delta_{\Phi}^{{}^{\prime}}F^{{}^{\prime}}$ implies that $E\times E^{{}^{\prime}}$ is descriptively near $F\times F^{{}^{\prime}}$, where $E\times E^{{}^{\prime}}$ and $F\times F^{{}^{\prime}}$ are any subsets of $X\times X^{{}^{\prime}}$.

In Definition 2.11, $G$ is a descriptive proximal homotopy between $h_{1}$ and $h_{2}$. We simply denote a descriptive proximal homotopy by “dprox-hom”. Given a descriptive proximity $\delta_{\Phi}$ on $X$ and a subset $F\subset X$, $F$ is said to be a $\delta_{\Phi}-$neighborhood of $E$, denoted by $E\ll_{\delta_{\Phi}}F$, if $E\underline{\delta_{\Phi}}(X-F)$[13].

In Definition 2.13, the fact $h:I\rightarrow X$ is descriptive proximally continuous means that “$D(E,F)=0\ \Rightarrow\ h(E)\delta_{\Phi}h(F)$” for $E$, $F\in I$. A descriptive proximity space $(X,\delta_{\Phi})$ is connected if and only if for all nonempty $E$, $F\in\mathcal{P}(X)$, $E\cup F=X$ implies that $E\delta_{\Phi}F$[7]. A descriptive proximity space $(X,\delta_{\Phi})$ is path-connected if, for any points $x_{1}$ and $x_{2}$ in $X$, there exists a descriptive proximal path from $x_{1}$ to $x_{2}$ in $X$.

The work of mapping spaces in nearness theory starts with [8] and is still open to improvement. Note that the study of discrete invariants of function spaces is essentially homotopy theory in algebraic topology, and recall that depending on the nature of the spaces, it may be useful to attempt to impose a topology on the space of continuous functions from one topological space to another. One of the best-known examples of this is the compact-open topology.

To show that the evaluation map $e_{X,Y}$ is a pc-map, we first assume that $(\{\alpha_{i}\}_{i\in I}\times E)\delta(\{\beta_{j}\}_{j\in J}\times F)$ in $Y^{X}\times X$. This means that $\{\alpha_{i}\}_{i\in I}\delta^{{}^{\prime}}\{\beta_{j}\}_{j\in J}$ for a proximity relation $\delta^{{}^{\prime}}$ on $Y^{X}$ and $E\delta_{1}F$ in $X$. It follows that $\alpha_{i}(E)\delta_{2}\beta_{j}(F)$ in $Y$ for any $i\in I$ and $j\in J$. Finally, we conclude that

Note that, in topological spaces, the map $X^{I}\rightarrow X\times X$, $\alpha\mapsto(\alpha(0),\alpha(1))$, is the path fibration. Similarly, the map $X^{I}\rightarrow X$, $\alpha\mapsto\alpha(0)$, is the path fibration with a fixed initial point at $t=0$.

A covering space of a topological space and the fundamental group are tightly related. One can categorize all the covering spaces of a topological space using the subgroups of its fundamental group. Covering spaces are not only useful in algebraic topology, but also in complex dynamics, geometric group theory, and the theory of Lie groups.

In Definition 3.10, $(X,\delta)$ is called a proximal covering space of $(X^{{}^{\prime}},\delta^{{}^{\prime}})$. For $i\in I$, $Y_{i}$ is said to be a proximal sheet. For any $x^{{}^{\prime}}\in X^{{}^{\prime}}$, $p^{-1}(\{x^{{}^{\prime}}\})$ is called a proximal fiber of $x^{{}^{\prime}}$. The map $p|_{Y_{i}}:Y_{i}\rightarrow Y^{{}^{\prime}}$ is a proximal isomorphism if the map $p:(X,\delta)\rightarrow(X^{{}^{\prime}},\delta^{{}^{\prime}})$ is a proximal isomorphism. However, the converse is not generally true. Given any proximity $\delta$ on $X$, it is obvious that the identity map on $X$ is always a proximal covering map.