In general topology, given a topological space $(X,\delta )$, it is natural to ask whether there is a metric for X such that $\delta$ is the metric topology. Such a metric metricizes the topological space and the space is said to be metrizable. Around the 1950s, through the efforts of R.H. Bing [1], Y.M. Smirnov and C.H. Dowker[2], J. Nagata [3], M.H. Stone [4], this problem mentioned was satisfactorily solved, and eventually, their comprehensive conclusion is called Nagata-Smirnov metrization admittedly in general topology, unquestionably, which is the most important theorem of topology. By that time, the main theory of topology had been perfected. However, scholars engaged in academic research never stopped exploring the unknown areas and sought new ways to gain a breakthrough in topological theory. In 1968, C.L. Chang [5] introduced the fuzzy set theory of Zadeh [6] into topology for the first time, which declared the birth of $[0,1]$-topology. Soon after that, J.A. Goguen [7] further generalized L-fuzzy set to the proposed $[0,1]$-topology and his theory has been recognized as L-topology nowadays. From then on, this kind of lattice-valued topology formed another important branch of topology, and thereafter many creative results and original thoughts were presented (see [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29], etc.).

Nevertheless, how to generalize classical metrics to the lattice-valued topology reasonably has always been a great challenge. So far, there are quite some fuzzy metrics introduced in the branch of learning (see [8,12,14,15,26,30,31,32], etc.). Considering the codomain is either an ordinary number or a fuzzy number, these metrics are roughly divided into two types.

One type is composed of these metrics, each of which is defined by such a function whose distance between objects is fuzzy, while the objects themselves are crisp. Additionally, each of them always induces a fuzzifying topology. In recent years, these metrics have been promoted by quite a few experts, such as I. Kramosil, J. Michalek, A. George, P. Veeramani, V. Gregori, S. Romaguera, J. Gutiérrez García, S. Morillas, F.G. Shi, etc. (see [15,16,25,33,34,35,36,37,38,39,40,41,42,43,44], etc.).

The other type consists of these metrics, each of which is defined by such a mapping $p:M\times M\to [0,+\infty )$, where M is the set of all standard fuzzy points of the underlying classical set X. In this case, every such fuzzy metric always induces a fuzzy topology (see [8,12,13,14,26,28], etc.).

About the latter, there are roughly three kinds of fuzzy metrics in history, with which the academic community has gradually been familiar. In addition, there is the fourth metric recently discovered. About the four fuzzy metrics, we will list them below one by one.

The first is the Erceg metric, which was presented in 1979 by M.A. Erceg [14]. Since then, many scholars have been engaged in this research and have obtained many beautiful results. Among them, a typical conclusion is J.H. Liang [23] showed Urysohn’s metrization theorem in 1984: an L-topological space is Erceg metrizable if it is $T_1$, regular, and $C_{II}$. In 1985, M.K. Luo [24] constructed an example of Erceg metric on $I^X$ whose metric topology has no $\sigma$-locally finite base, which implies that the $[0,1]$-topological space of this example is not $C_{II}$ of course, but then Liang’s this conclusion is still the best one. In this paper, Liang’s conclusion is only a corollary of our result in $[0,1]$-topology. Later on, based on Peng’s simplification method [45], the Erceg metric was further simplified by P. Chen and F.G. Shi (see [11,46]) below:

(I) An Erceg pseudo-metric on $L^X$ is a mapping $p:M\times M\to [0,+\infty )$ satisfying

where $``\ll "$ is the way-below relation in Domain Theory and $L^X$ is a completely distributive lattice [47,48].

The second is the Yang-Shi metric (or p.q. metric), which is proposed in 1988 by L.C. Yang [28]. It was proved by Yang that each topological molecular lattice with $C_{II}$ property is p.q. metrizable (refer to [28,48] for details). After that, this kind of metric was studied in depth by F.G. Shi (see [11,26,46,49,50], etc.). Its definition is as follows:

(II) A Yang-Shi pseudo-metric (resp., Yang-Shi metric) on $L^X$ is a mapping $p:M\times M\to [0,+\infty )$ satisfying (A1)-(A3) (resp., (A1)-(A4)) and the following

Similarly, according to our later proofs in this paper, Yang’s this conclusion is still a corollary of our result in $[0,1]$-topology.

The third is the Deng metric supplied in 1982 by Z.K. Deng [12], where Deng [13] showed that if a $[0,1]$-topological space is $T_1$, regular and $C_{II}$ then it is Deng metrizable. In this paper, we will extend this result substantially. Incidentally, Y.Y. Lan and F. Long also provided a result about Deng pseudo-metrization problem [51]. However, the proof was not completely right after careful checking pointed out by us. It is worth mentioning that since Deng’s research is only limited to this special lattice $I^X$, Deng pseudo-metric was later extended to $L^X$ by P. Chen [52] below:

(III) An extended Deng pseudo-metric (resp., extended Deng metric) on $L^X$ is a mapping $p:M\times M\to [0,+\infty )$ satisfying (A1)-(A3) (resp., (A1)-(A4)) and the following

In a summary, the above three kinds of fuzzy metrics are defined by using (A1)-(A4) and different (B1), (B2), and (B3), respectively. Inspired by this, we conclude that there is another new metric defined as follows:

(IV) A Chen pseudo-metric (resp., Chen metric) on $L^X$ is a mapping $p:M\times M\to [0,+\infty )$ satisfying (A1)-(A3) (resp., (A1)-(A4)) and the following

About it, some elementary properties related to it have been introduced [8].

In this paper, we will focus on the latter mainly and study its metrization problem in $[0,1]$-topology. For this reason, we investigate the relationships between (I)-(IV) on $I^X$ and fortunately acquire such a profound result: let $C=\{$$p\mid p$ is a Chen metric}, $E=\{$$p\mid p$ is an Erceg metric }, $D=\{$$p\mid p$ is a Deng metric } and $Y=\{$$p\mid p$ is a Yang-Shi metric } on $I^X$. Then $D=C\cap Y\cap E$.

Consequently, if a given $[0,1]$-topology is Deng metrizable, then it must be Erceg, Yang-Shi, and Chen metrizable. Thus this paper mainly will discuss the Deng metric and its metrization problem in $[0,1]$-topology.

To sum up, although so many scholars have been engaged in the study of fuzzy metrics, it is a little pity that the metrization problem in $[0,1]$-topology remains unsolved now. this paper aims to study the metrization problem in $[0,1]$-topology and will obtain the generalization of Nagata–Smirnov metrization theorem in $[0,1]$-topology.

In this section, we cite the fundamental definitions that will be used in the sequel. The letter X always refers to a nonempty set throughout this paper, and I denotes the unit interval $[0,1]$.

A fuzzy set of X is a mapping $A:X\to I$, which forms the family $I^X$. The constant fuzzy set of X with the value 1 (resp., 0) is denoted by $\underline{1}$ (resp., $\underline{0}$). A fuzzy point (resp., standard fuzzy point) $x_{\lambda }$ in X is a fuzzy set defined by $x_{\lambda }\left(x\right)=\lambda$ and $x_{\lambda }\left(y\right)=0$ if $y\ne x$, where $\lambda$ is a fixed number in $(0,1)$ (resp., $(0,1]$). The set of all fuzzy points (resp., all standard fuzzy points) of X is denoted by $M_0$ (resp., M). $M_0$ is a subfamily of M. Naturally, these properties of $M_0$ and M are also suitable for L-topology.

A subfamily $\delta$ of $I^X$ is called a $[0,1]$-topology if it satisfies the following three conditions: (O1) $\underline{1},\underline{0}\in \delta$; (O2) if $A,B\in \delta$, then $A\wedge B\in \delta$; (O3) if $\{A_{\lambda }\mid \lambda \in \Lambda \}\subseteq \delta$, then $\underset{\lambda \in \Lambda }{\bigvee }{A}_{\lambda }\in \delta$. The pair $(X,\delta )$ is called a $[0,1]$-topological space (a space for short). If $\delta \subseteq I^X$, then for each $A\in \delta$, A and $A^{\prime }$ are called a $\delta$-fuzzy open set and a $\delta$-fuzzy closed set (open set and closed set for short), respectively.

Two fuzzy sets A and B are called quasi-coincident if there exists x belonging to X such that $A\left(x\right)+B\left(x\right)>1$ (see [53]). Let $x_{\alpha }$ be a fuzzy point and let A be a fuzzy set of X. The notation $x_{\alpha }\in A$ means $\alpha <A\left(x\right)$ [12]. The closure of a fuzzy set A of $(X,\delta )$ is the intersection of the members of the family of all closed sets containing A, denoted by $\overline{A}$ [12]. A fuzzy point $x_{\alpha }$ is called a cluster point of a fuzzy set U of $(X,\delta )$ if each open neighborhood of $x_{1-\alpha }$ is quasi-coincident with U. Consequently, $x_{\alpha }\le \overline{A}$ if and only if $x_{\alpha }$ is a cluster point of A. Therefore, $\overline{A}=\bigvee \{y_{\beta }\mid y_{\beta }$ is a cluster point of $\overline{A}\}$ [12].

The space $(X,\delta )$ is called regular (resp., normal) if for any $x_{\lambda }\in M_0$ (resp., ${\tau }^{\prime }\in \delta$), $\mu \in \delta$ with $x_{\lambda }\in \mu$ (resp., $\tau \le \mu$), there exists $\upsilon$ belonging to $\delta$ such that $x_{\lambda }\in \upsilon \le \overline{\upsilon }\le \mu$ (resp., $\tau \le \upsilon \le \overline{\upsilon }\le \mu$) [20]. A $[0,1]$-topological space is $T_1$ if and only if $x_{\lambda }$ is closed for each fuzzy point $x_{\lambda }\in M_0$. A family $\psi$ of fuzzy sets is a base of $\delta$ if $\psi$ is a subfamily of $\delta$ and for each fuzzy point $x_{\lambda }$ and each open neighborhood $\mu$ of $x_{\lambda }$, there is a member $\upsilon$ of $\delta$ such that $x_{\lambda }\in \upsilon \le \mu$. A family $\kappa$ of fuzzy sets is a subbase of $\delta$ if the family of finite intersections of members of $\kappa$ is a base of $\delta$ [12,54]. The space $(X,\delta )$ is $C_{II}$, or called second-countable if the $[0,1]$-topology $\delta$ has a countable basis.

A family of fuzzy sets $\Psi$ is called locally finite (resp., discrete) in a space $(X,\delta )$ if and only if each fuzzy point $x_{\lambda }$ of the space has its an open neighborhood which is quasi-coincident with only finitely many members (resp., at most one member) of $\Psi$ (see [48]). A family of fuzzy sets is called $\sigma$-locally finite (resp., $\sigma$-discrete) in a space $(X,\delta )$ if and only if it is the union of a countable number of locally finite (resp., discrete) subfamilies. A subfamily $\sigma$ of $I^X$ (resp., $\sigma$ of $\delta$) is called a (resp., an open) cover of a fuzzy set A in a space $(X,\delta )$ if for each $x_{\alpha }\in A$, there exists B belonging to $\sigma$ such that $x_{\alpha }\in B$. Furthermore, if $A=\underline{1}$, then $\sigma$ is called a cover of $(X,\delta )$. A cover $\mathcal{B}$ of a fuzzy set A is called a refinement of a cover $\mathcal{D}$ if each member of $\mathcal{B}$ is a subset of a member of $\mathcal{D}$ [48].

Let ${\left\{X_t\right\}}_{t\in T}$ be an indexed family of sets. The cartesian product of this indexed family, denoted by $\prod _{t\in T}{X}_t$, is the set of all functions $x:T\to \bigcup _{t\in T}{X}_t$ such that $x\left(t\right)\in X_t$ for each $t\in T$.

Let $X=\prod _{t\in T}{X}_t$. Then the t-th projection$J_t:I^X\to I^{X_t}$ is defined by $J_t\left(A\right)\left(y_t\right)=sup\{A\left(x\right)\mid x_t=y_t\}foreach{y}_t\in X_t$ and let $J_t^{-1}\left(B\right)=\bigvee \{C\in I^X\mid J_t\left(C\right)\le B\}$. The product space of $\{(X_t,{\delta }_t)\mid t\in T\}$ is defined by $\{J_t^{-1}\left(A_t\right)\mid A_t\in {\delta }_t,t\in T\}$ as a sub base [48].

Other unexplained terminologies and notations and further details can be found in [7,8,12,48,54].

Remark In [52] we have proved the following results: (1) a Deng metric p on $I^X$ can be extended to an extended Deng metric $p^{*}$; (2) $p=p^{*}\mid M_0\times M_0$; (3) $p^{*}$ and p induce the same metric topology.

Based on the above (1)-(3), it is much easier to study Deng metric by using Definition 1 instead of $\left(\mathbf{III}\right)$ on $I^X$ as its definition. A similar treatment to $\left(\mathbf{I}\right)$, $\left(\mathbf{II}\right)$, and $\left(\mathbf{VI}\right)$ on $I^X$ is to restrict their domains to $M_0$ and use (D4) instead of (A3) while other conditions remain unchanged.

In [12], Deng has proved such a result: If $(X,\delta )$ is regular and $C_{II}$, then it is normal [12]. It is a special case of the following result:

In this section, we will investigate the relationships between the four kinds of metrics: Erceg , Yang-Shi, Deng, and Chen metrics. First of all, we expose the main result as follows:

However, the converse is not true. Such a counterexample is given below.

Firstly, let us prove that p is an Erceg pseudo-metric on $[0,1]$.

(A1) and (A2) are trivial.

(B1) if $a,b\in (0,1]$ and $a\ge b$, then $p(a,b)=0$. Therefore, we have $\underset{x<b}{\bigvee }p(a,x)=0$, so that in this case $p(a,b)=\underset{x<b}{\bigvee }p(a,x)$. Similarly, when $a<b$, we can prove $p(a,b)=\underset{x<b}{\bigvee }p(a,x)=1$. Consequently, p satisfies (B1).

(A3) we only need to prove that $\underset{y>1-b}{\bigwedge }p(a,y)=\underset{x>1-a}{\bigwedge }p(b,x)$, which can be obtained from the following implications: $\underset{y>1-b}{\bigwedge }p(a,y)=1\iff y>1-b$ implies $y>a\iff 1-b\ge a\iff x>1-a$ implies $x>b\iff \underset{x>1-a}{\bigwedge }p(b,x)=1$.

Secondly, we assert that p is not a Yang-Shi pseudo-metric. In fact, for any $a\in (0,1]$, we have $p(a,a)=0$. But $\underset{c<a}{\bigwedge }p(c,a)=1$. Thus $p(a,b)\ne \underset{c<a}{\bigwedge }p(c,b)$, as desired. □

Conversely, we have the following conclusion:

To prove this, we first need to prove the following two Lemma 1 and Lemma 2.

Conversely, we have the following result:

In summary, because of Theorem 8 in this section, we have asserted that if a given $[0,1]$-topology $\delta$ is Deng metrizable, then $\delta$ must be Erceg, Yang-Shi, and Chen metrizable. For this reason, next, we will mainly focus on the Deng metric and its metrization in $[0,1]$-topology.

In this section, let $Q_{[0,1]}$ be the set of all rational numbers in $[0,1]$, let $\omega =\{1,2,\cdots ,n,\cdots \}$, and let $\mathcal{X}=\prod _{n\in \omega }{X}_n$, where $X_n$ ($n\in \omega$) is a nonempty set. For clarity, the set of all fuzzy points in $\mathcal{X}$ is denoted by $M_0\left(\mathcal{X}\right)$, and for $y\in \mathcal{X}$, $y=(y^1,y^2,\cdots ,y^n,\cdots )$.