1 Introduction

Zadeh added the essential standards of fuzzy sets in his classical paper [14]. Fuzzy sets have applications in lots of fields of engineering, social technological know-how, economics, clinical science and many others,. In mathematics, topology furnished the most natural framework for the ideas of fuzzy units to flourish. Bayramov [1] added and advanced the concept of L-fuzzy topological spaces. Additionally fibrewise variations of homotopy concept were studied in [8, 11]. The perception of fuzzy homotopy principle was delivered by way of G. Culvacioglu and M. Citil in [10]. The essential organization of fuzzy topological areas was brought by using Abdul Razak Salleh and Mohammad faucet in [2, 7]. Prompted through [2, 7], fuzzy essential organization in fuzzy topological areas became prolonged to numerous fuzzy structure spaces in [5, 6]. The concept of compact-open topology has a important position in defining function spaces in standard topology. Routaray et al. [9] introduced the concept of fuzzy - structure covering map, fuzzy \(\Im ^{*}\)-Structure compact open topology.

One of the most crucial ideas in general topology is compactness, which many authors [3, 13] have extended to L-topological space.The objective of this article is to present an original concept of L-fuzzy compact-open topology and contribute some theories and effects relating to this concept.

The rest of paper is organized as follows: Section "Introduction" highlights the importance of fuzzy topology. The "Preliminaries" section emphasizes the fuzzy logic and L-fuzzy topology. The next part discusses the L-fuzzy covering space and some related theorems. The section "L-fuzzy compact open topology" details suggested the concept of L-fuzzy compact open topology, L-fuzzy connected set and L-fuzzy path connected space. The contributions of the work are summarized in the "Conclusion" section which highlights the achievements and identifies possible direction for further study.

2 Preliminaries

Definition 2.1

[14] A function from a non-empty set X to a unit interval \(I = [0, 1]\) is called a fuzzy set \(\lambda \). \( I^{X} \) denotes the fuzzy set family as a whole.

Definition 2.2

[8] Let X be a set and \( \tau \) be a family of fuzzy subsets of X. Then \( \tau \) is called fuzzy topology on X if satisfies the following conditions:

(i):

\(0_{X}, 1_{X}\in \tau \)

(ii):

If \( \lambda , \mu \in \tau \) then \( \lambda \wedge \mu \in \tau \)

(iii):

If \( \lambda _{i}\in \tau \) for all I then \( \vee \lambda _ {i}\in \tau . \)

Definition 2.3

[12] For any fuzzy set \(A \in F (X)\) and any \(\lambda \in [0, 1]\), the \(\lambda \)-cut and strong \(\lambda \)-cut of A are respectively defined as follows: \(A_{\lambda } =\left\{ x \in X: A (x) \ge \lambda \right\} , A_{\left\langle \lambda \right\rangle }=\left\{ x\in X:A(x)\ge \lambda \right\} \), where \(A(x) = \mu A (x)\) since A(x) is more convenient than \(\mu A (x)\).

Definition 2.4

[12] Let \(I^{\tau }\) be set of all monotonic decreasing maps \(\lambda :\mathbb {R}\rightarrow L \) (where L is completely distributive lattice) satisfying:

(i):

\(\lambda (t) = 1\) for \(t < 0\),

(ii):

\(\lambda (t) = 0\) for \(t > 1\).

For \(\lambda , \mu \in I^{\Gamma }\), we define that \(\lambda \equiv \mu \) iff \(\lambda (t-) = \mu (t-)\) and \(\lambda (t+) = \mu (t+)\) for all \(t \in \mathbb {R},\) where \(\lambda (t-) = \text {inf}_{s< t}\lambda (s)\) and \(\lambda (t+)=\text {sup}_{s>t}\lambda (s)\). Then \(\equiv \) is an equivalence relation on \(I^{\Gamma }, \left[ \lambda \right] \) denotes the equivalence class of \(\lambda \in I^{\Gamma }\) and the quotient set \(I^{\Gamma } / \equiv \) is called the L- fuzzy unit interval which in symbols is written I(L).

We define an L-fuzzy topology \(\tau \) on I(L) by taking as a subbase \(\left\{ L_{t, R_{t}:t\in \mathbb {R}} \right\} \), where we define \(L_{t}\left( \left[ \lambda \right] \right) = {(\lambda (t-))}'\) and \(R_{t}\left( \left[ \lambda \right] \right) = {(\lambda (t+))}'\). The topology \(\tau \) is called the standart topology on I(L), and the base of \(\tau \) is \(\left\{ L_{s}\wedge R_{t}: s, t\in \mathbb {R} \right\} \).

Definition 2.5

Let \( f,g:(X,\tau )\rightarrow (Y,\sigma ) \) be L-fuzzy continuous maps. We say that f is L-fuzzy homotopic to g if there exists an L-fuzzy continuous map \(F:(X,\tau )\times (I(L),\tau )\rightarrow (Y, \sigma )\) such that \(F(a_{\alpha },[\lambda _{0}])=f(a_{\alpha })\) and \(F(a_{\alpha },[\lambda _{1}])=g(a_{\alpha })\) for every L-fuzzy point \(a_{\alpha } \in (X, \tau )\) where \(i=0,1.\)

$$\begin{aligned} \lambda _{i}(t)=\left\{ \begin{matrix} 1, &{} t<i\\ 0, &{} t>i \end{matrix}\right. \end{aligned}$$

The map F is called an L-fuzzy homotopy between f and g, and written \(F:f\cong _{L}g\).

Definition 2.6

Let \( (X,\tau ) \) and \( (Y, \sigma ) \) be any two L-fuzzy space. Let \( p:(X,\tau )\rightarrow (Y,\sigma ) \) is called a L-fuzzy covering space if and only if

(i):

p is L-fuzzy onto.

(ii):

For every \( a_{\alpha } \in X\) there exists a neighborhood \( a_{\alpha }\in U\) such that \(p^{-1}(u)=\cup S_{i}\) such that each \(s_{i}\) is L-fuzzy homeomorphic to U.

Each \(s_{i}\) is called a sheet. Each u for which \(p^{-1}(u)=\cup s_{i}\) is said to be L-fuzzy covered. \(p^{-1}(a_{\alpha })\) is called a L-fuzzy fiber.

Definition 2.7

Let \(p: (X, \tau )\rightarrow (Y,\sigma )\) be a L-fuzzy covering map and let \(f: (\tilde{X},\tilde{\sigma })\rightarrow (Y,\sigma )\) be a L-fuzzy continuous function. Then a map \( \tilde{f}: (\tilde{X},\tilde{\sigma })\rightarrow (X,\tau )\) is said to be L-fuzzy lift on the map f if \( p\circ \tilde{f}=f \).

3 L-Fuzzy covering space

Theorem 3.1

Let \(p: (X_{1},\tau , x_{\lambda })\rightarrow (X_{2},\sigma ,x_{\lambda _{2}}) \) be a L-fuzzy covering space generated by \( \tau \) and \( \sigma \). Let \( f:(Y,\eta , y_{\lambda })\rightarrow (X_{2},\sigma ,x_{\lambda _{2}}) \) be a arbitrary L-fuzzy map. If \( (Y, \eta , y_{\lambda }) \) is L-fuzzy connected then \( f^{'} \) is unique (if it exists).

Proof

Let \( f^{''} \) be another L-fuzzy lifting of the map \( pf^{'}=f, pf^{''}=f \). Define

$$\begin{aligned} A=\left\{ y\in Y: f^{'}(y)=f^{''}(y) \right\} \end{aligned}$$

and

$$\begin{aligned} B=\left\{ y\in Y: f^{'}(y)\ne f^{''}(y) \right\} \end{aligned}$$

clearly \(Y=A\cup B\) and \(A\cap B=\phi .\)

For \( y\in A \) we have \( f^{'}(y)=f^{''}(y)\) and \(pf^{'}(y)=f(y)\) implies \(pf^{'}(y)\in u\) from this line it is clear that \(f^{'}(y)\in p^{-1}(u). \) Again \( f^{'}(y)\in S, f^{''}(y)\in S\) this implies \(y\in f^{'}(S) \cap f^{''}(S)\subset A\).

If not let there exists \(z\in B\) so \(f^{'}(z)\in S, f^{''}(z)\in S\) and \(f^{'}(z)\ne f^{''}(z)\) implies \(pf^{'}(z)\ne pf^{''}(z)\). So we get \(f(z)\ne f(z)\) which is a contradiction. So A is a L-fuzzy open set. Similarly B is also L-fuzzy open. Since Y is L-fuzzy connected one of A and B must be empty.

Clearly \( f^{'}(x_{\lambda }) =y_{\lambda }\) and \( f^{''}(x_{\lambda }) =y_{\lambda }\). So \( x_{\lambda }\in A \) and \( A\ne \phi , B=\phi . \) This competes the proof. \(\square \)

Theorem 3.2

Let \(p: (X_{1},\tau , x_{\lambda })\rightarrow (X_{2},\sigma , x_{\lambda _{2} }) \) be a L-fuzzy covering space and \( \omega \) be a L-fuzzy path in \((X_{1},\tau , x_{\lambda })\), then there exists a unique \( \omega ^{'}:I\rightarrow X_{1} \) such that \( \omega ^{'}=\omega \).

Proof

Since I is L-fuzzy connected, \( \omega ^{'} \) (if it exists) must be unique. Now we shall prove that \( \omega ^{'} \) exists.

Case-1: Suppose \( X_{2} \) itself is L-fuzzy structure covered, i.e., \(p^{-1}(X_{2})=\cup S_{i} =E\). \(x_{\lambda _{1} }\) belongs to some sheet \(S_{i}\). Then \(p/S_{i}\) is a homeomorphism and \(\psi \) be the inverse map, i.e., \(\psi :X_{2}\rightarrow S_{i}\). Clearly \(\psi \) exists and L-fuzzy continuous. Since \(p/S_{i}\) is a homeomorphism. Let \(\omega :I\rightarrow X_{2}\) then \(\psi \circ \omega :I\rightarrow S_{i}\), i.e., E. Let \(\omega ^{'} =\psi \circ \omega \). So \(\omega ^{'}\) is a L-fuzzy path in \(X_{2}\). This is our required \(\omega ^{'}\) is a L-fuzzy path in \(S_{i}\).

To show that \( p\omega ^{'}=\omega \) \(p\omega ^{'}=(p/S_{i})\omega ^{'}=(p/S_{i})\psi \circ \omega =\omega \) since \((p/S_{i})\psi =id\), i.e., \(\psi \) is the inverse of \(p/S_{i}\).

Case-II: For each \( x_{\lambda }\in X_{2} \) there exists a L-fuzzy neighborhood \( x_{\lambda }\in X_{U_{\lambda }} \) which is L-fuzzy covered and each \( \left\{ \omega ^{-1} (U_{x_{\lambda }}) \right\} \) is a L-fuzzy open set. Thus the collection of these L-fuzzy open sets will be a L-fuzzy covering I. Since I is L-fuzzy compact it is possible to choose a L-fuzzy finite covering

$$\begin{aligned} 0=t_{0}< t_{1}< t_{2}\cdots < t_{n}=1 \end{aligned}$$

such that \( [t_{i},t_{i+1}]\subset \omega ^{-1}(U_{x})\) for some x and for all i. \(\omega [t_{i},t_{i+1}]\subset \omega ^{-1}(U_{x}) \) for some x and for all i.

There exists \(\omega _{1}^{'}:[t_{0},t_{1}]\rightarrow X_{1}\) such that \(p\omega _{1}^{'}=\omega /[t_{0},t_{1}]\) similarly there exists \(\omega _{2}^{'}:[t_{1},t_{2}]\rightarrow X_{1}\) such that \(p\omega _{2}^{'}=\omega /[t_{1},t_{2}]\)

Define \(\omega ^{'}(t)=\left\{ \begin{matrix} \omega _{1}^{'}(t) \;\;\; t_{0}\le t\le t_{1}&{} \\ \omega _{2}^{'}(t) \;\;\; t_{1}\le t\le t_{2} &{} \\ \vdots &{} \\ \omega _{n}^{'}(t) \;\;\; t_{n-1}\le t\le t_{n} \end{matrix}\right. \) \(\square \)

Theorem 3.3

Let \(f:(Y,\tau )\rightarrow (X,\sigma )\) admit a L-fuzzy lifting \(f^{'}\) generated by \( \tau \) and \(\sigma \); then any L-fuzzy homotopy \(F:(Y,\tau )\times (I(L),\tau )\rightarrow (X,\sigma )\) with \(F(a_{\alpha },[\lambda _{0}])=f(a_{\alpha })\) can be L-fuzzy lifted to a L-fuzzy homotopy \(F^{'}:(Y,\tau )\times (I(L),\tau )\rightarrow E\) with \(F^{'}(a_{\alpha },[\lambda _{0}])=f^{'}(y)\)

Proof

Given that \( F(a_{\alpha },[\lambda _{0}])=f(a_{\alpha }).\) Theorem states that there exists \( F^{'} \) such that \( F^{'}(a_{\alpha },[\lambda _{0}])=f^{'}(a_{\alpha }) \).

Case-1: If the whole space X is evenly L-fuzzy covered then \(p^{-1}(X)=\cup S_{i}=E.\) \(p/S_{i}\) defines a L-fuzzy homeomorphism from \(S_{i}\) to X. Thus \(S_{i}\cong X\). Hence there exists inverse map \(\psi \) such that \(F^{'}=\psi \circ F\) where \(F^{'}:Y\times I\rightarrow E\). Clearly \(F, \psi \) and \(F^{'}\) are L-fuzzy continuous. \(F^{'}(a_{\alpha },[\lambda _{0}])=\psi F(a_{\alpha },[\lambda _{0}])=\psi f(a_{\alpha })=f^{'}(a_{\alpha }) \).

Case-II: Let \(a_{\alpha }\in Y\) and \(\lambda _{t}\in I\), then \(F(\lambda _{t},a_{\alpha })\in X\). Since E is L-fuzzy covering space there exists a L-fuzzy neighborhood \(U_{F(a_{\alpha },\lambda _{t})}\) which is evenly covered \(\left\{ F^{-1} (U_{F(a_{\alpha },\lambda _{t})})\right\} \) where \(a_{\alpha }\) is fixed and \(\lambda _{t}\in I\).Clearly \( \lbrace a_{\alpha } \rbrace \otimes I\)is L-fuzzy compact set and \(F^{-1} (U_{F(a_{\alpha },\lambda _{t})}) \) is a L-fuzzy covering set of \( \lbrace a_{\alpha } \rbrace \otimes I\). Now consider \( 0=\lambda _{t_{0}}< \lambda _{t_{1}}< \lambda _{t_{2}}< \cdots \lambda _{t_{n}}=1\) and we get \(\left\{ a_{\alpha } \right\} \times [\lambda _{t_{i}},\lambda _{t_{i+1}}]\subset F^{-1}(U_{F(a_{\alpha },\lambda _{t})})\). But \(F^{-1}(U_{F(a_{\alpha },\lambda _{t})})\) is a L-fuzzy open set containing \((a_{\alpha },\lambda _{t})\). Choose a neighbourhood \(N_{a_{\alpha }}\) of \(a_{\alpha }\) such that \(F(N_{1}\times [\lambda _{t_{i}},\lambda _{t_{i+1}}])\subset U_{F(a_{\alpha },\lambda _{t})}\). Thus \(U_{F(a_{\alpha },\lambda _{t})}\) is evenly L-fuzzy covered. \(\square \)

Corollary 3.4

Let \((X, \tau )\) and \( (Y, \sigma )\) be two L-fuzzy structure and \((I(L), \tau )\) be L-fuzzy space introduced by \(\tau \). Let \(\alpha , \beta :I(L)\rightarrow X\) and \(p:Y\rightarrow X\).If \(\alpha \simeq _{L}\beta \) then \( \alpha ^{'}\simeq _{L} \beta ^{'}\).

Proof

Let \(\alpha : (I(L), \tau ) \rightarrow (X,\tau )\) be any \( L- \)fuzzy continuous function. Define a function \(F: (I(L), \tau ) \times (I(L), \tau ) \rightarrow (X, \tau )\) such that \( F(a_{\alpha },[\lambda _{0}])=\alpha (a_{\alpha })\) and \( F(a_{\alpha }, [\lambda _{1}])=\beta (a_{\alpha }).\) Then there exists \( F^{'} \) such that \( F^{'}(a_{\alpha },[\lambda _{0}])=\alpha ^{'}(a_{\alpha }) \), \( pF^{'}(a_{\alpha },[\lambda _{1}])=F(a_{\alpha },[\lambda _{1}]) \) and \(pF^{'}(a_{\alpha },[\lambda _{1}])=\beta (a_{\alpha }).\) So\( F^{'}(a_{\alpha },[\lambda _{1}]) \) is a lifting of \( \beta \). \( \beta ^{'} \) is also lifting of \( \beta \). Since I(L) is L-connected both are equal.\( \beta ^{'}(a_{\alpha })=F^{'}(a_{\alpha },[\lambda _{1}]) \). Clearly we have \( \alpha ^{'}\simeq _{L}\beta ^{'} \). \(\square \)

Corollary 3.5

Let \((X, \tau )\) and \((Y, \sigma )\) be any two L-fuzzy path connected spaces and \(p: (X, \tau ) \rightarrow (Y, \sigma )\) be a L-fuzzy continuous function. Let \(x_{\lambda }\) be a fuzzy point in \((X, \tau )\) and \( p_{*}: \Pi _{1}(X, x_{\lambda }) \rightarrow \Pi _{1}(Y, p(x_{\lambda })).\) Then \(p_{*}\) induces a L-fuzzy monomorphism.

Proof

For each \([\alpha ], [\beta ] \in \Pi _{1}(X, x_{\lambda })\),

$$\begin{aligned} p_{*}([\alpha ] \circ [\beta ])&= p_{*} [\alpha *\beta ]\\&=[p \circ (\alpha *\beta )]\\&=[(p \circ \alpha ) *(p \circ \beta )]\\&=[p \circ \alpha ] *[p \circ \beta ]\\&==p_{*} ([\alpha ]) \circ p_{*}([\beta ]). \end{aligned}$$

Thus \(p_{*}\) is a L-fuzzy homomorphism. Let \([\alpha ]\in \Pi _{1}(X, x_{\lambda })\) then \( p_{*}[\alpha ]=[\gamma ]\) (where \(\gamma :I\rightarrow Y\)) this implies \([p\circ \alpha ]\simeq [\gamma ]\). So we get \(p\circ \alpha \simeq _{L} \gamma \). Again \(\gamma \) has a lifting in X. Suppose \(\delta \) is its lifting, i.e., \(p\delta =\gamma \). \( p\delta (x_{t})=\gamma (x_{t})=p(x_{t})\). Let \((x_{t})=y_{t}\). Now we get \(\delta (x_{t})\in p^{-1}(y_{t})\). But \(p^{-1}(y_{t})\) is a fuzzy discrete set of points in X. \(\delta (I)\subset p^{-1}(y_{t})\), \(\delta (I)\) is L-fuzzy connected, hence it must a singleton. Since the points are L-fuzzy discrete \(\delta (I)=x_{\lambda }. \delta \) is the L-fuzzy lifting of \(\gamma \) and \(\alpha \) is the L-fuzzy lifting of \(p\circ \alpha \). Therefore \(\alpha \simeq _{L} \delta \) and \([\alpha ] \) is the identity class. \(\square \)

4 L-Fuzzy compact open topology

Let \((X,\tau )\) and \( (Y,\sigma ) \) be any two L-fuzzy spaces generated by \(\tau \) and \( \sigma \).

Let

$$\begin{aligned} Y^{X}=\lbrace f:(X,\tau )\rightarrow (Y,\sigma )|\;\;f\; \text {is}\; L\text {-fuzzy}\; \text {continuous function}.\rbrace \end{aligned}$$

We give this class \( Y^{X} \) a topology called the L-fuzzy compact open topology as follows:

Let

$$\begin{aligned}{} & {} \kappa :\lbrace K:I\rightarrow X: K \;\text {is}\; L\text {-fuzzy compact}\; \text {in}\; X\rbrace .\\{} & {} \eta =\left\{ U:I\rightarrow Y \;\text {such that }\;U \;\text {is}\; L\text {-fuzzy}\; \text {open in}\; Y \right\} . \end{aligned}$$

For any \(K\in \kappa \) and \(U\in \eta \), let

$$\begin{aligned} W(K,U)=\left\{ \omega \in Y^{X}:\omega (K)\subseteq U \right\} . \end{aligned}$$

The collection \(\left\{ W(K,U):K\in \kappa , U\in \eta \right\} \) can be as a fuzzy subbase to generate a L-fuzzy topology on the class \( Y^{X} \), called the L-fuzzy compact-open topology. The class \( Y^{X} \) with this topology is called L-fuzzy compact-open topological space. Unless otherwise stated, \( Y^{X} \) will always have the L-fuzzy compact-open topology.

Theorem 4.1

Let \((X, \tau )\) and \( (Y, \sigma )\) be two L-fuzzy compact space. Let \(a_{\alpha }\) be any L-fuzzy point in X and N be a L-fuzzy open set in the L-fuzzy product space \(X \times Y\) containing \({a_{\alpha }} \times Y\). Then there exists some L-fuzzy neighborhood W of \(a_{\alpha }\) in X such that \( a_{\alpha }\times X\subseteq W\times Y\subseteq N \).

Proof

It is clear that \(x_{t} \times Y\) is L-fuzzy homeomorphic to Y and hence \({x_{t}} \times Y\) is L-fuzzy compact. We cover \(\lbrace x_{t}\rbrace \times Y\) by the basis elements \(\lbrace U \times V\rbrace \) (for the L-fuzzy topology of \(X \times Y\)) lying in N. Since \(\lbrace x_{t}\rbrace \times Y \) is L-fuzzy compact, \(\lbrace U \times V\rbrace \) has a finite subcover, say, a finite number of L-fuzzy basis elements \( U_{1}\times V_{1},\cdots U_{n}\times V_{n} \). Without loss of generality we assume that \( x_{t}\in U_{i} \) for each \(i= 1,2 \cdots n\); since otherwise the basis elements would be superfluous. Let \( W=\bigcap _{i=1}^{n}U_{i} \). Clearly W is L-fuzzy open and \(x_{t}\in W\). We show that

$$\begin{aligned} W\times Y\subseteq \bigcup _{i=1}^{n}(U_{i}\times V_{i}) \end{aligned}$$

Let \( (x_{r},y_{s}) \) be any L-fuzzy point in \(W \times Y\). We consider the L-fuzzy point \((x_{t},y_{s})\). Now \( (x_{t},y_{s})\in U_{i}\times V_{i} \) for some i; thus \(y_{s}\in V_{i}\). But \( x_{r}\in U_{j} \) for every \(j = 1, 2 \cdots n \) (because \( x_{r}\in W \)). Therefore \( (x_{r},y_{s})\in U_{i}\times V_{i} \), as desired. But \( U_{i}\times V_{i}\subseteq N \)for all \(i = 1, 2, \cdots , n\); and \( W \times Y \subseteq \bigcup _{i=1}^{n}(U_{i}\times V_{i}). \) Therefore \(W \times Y \subseteq N\). \(\square \)

Theorem 4.2

Let Y be a L-fuzzy connected and L-fuzzy locally path connected space. \( P(Y,y_{0})\) denote the set of all L-fuzzy paths whose initial point is \( y_{0} \). Then \( \phi :P(Y,y_{0})\rightarrow Y \) is L-fuzzy continuous, L-fuzzy onto and L-fuzzy open map.

Proof

We have

$$\begin{aligned} W(K,U)=\left\{ \omega \in Y^{X}:\omega (K)\subseteq U \right\} . \end{aligned}$$

So clearly \( \left\{ \omega :\omega (1)\subset U \right\} \) be L-fuzzy open set in the L-fuzzy compact open topology. Let \(\phi :P(Y,y_{0})\rightarrow Y \) and consider u as L-fuzzy open set in Y. So

$$\begin{aligned} \phi ^{-1}(u)=\left\{ \omega :\phi (\omega )\in u \right\} =\left\{ \omega :\omega (1)\in u \right\} \end{aligned}$$

is L-fuzzy open. This implies \( \phi \) is L-fuzzy continuous map.

Let \(\omega \in P(Y,y_{0})\) and \(W=\bigcap _{i=1}^{n}W(K_{i},U_{i})\) where \(K_{i}\) be L-fuzzy compact and \(U_{i}\) be L-fuzzy open set. Arrange the \( K_{i} \) according to decreasing end points. Choose \( j\le n \) such that

$$\begin{aligned} 1\in K_{1}\cap K_{2}\cap \cdots K_{j} \end{aligned}$$

and

$$\begin{aligned} 1\notin K_{j+1}\cap K_{J+2}\cap \cdots K_{n} \end{aligned}$$

Again \(\omega \in W(K_{i},U_{i})\) then \(\omega (K_{i})\subset U_{i}\) for all \(i=1,2,\cdots n\) and \(i=1,2,\cdots j\). From this it is clear that \( \omega (1)\in \bigcap _{i=1}^{n}U_{i} \). Choose L-fuzzy connected neighborhood of V of \( \omega (1)\) such that

$$\begin{aligned} V\subset \cap _{i=1}^n U_i. \end{aligned}$$

Choose \(t'\in (0,1)\) such that

$$\begin{aligned} {[}t',1]\cap [K_{j+1}\cup ~~.~~.~~.~~\cup K_{n}]=\phi \end{aligned}$$

such that \(\omega [t',1]\subset V\).

We claim that \( V\subset \phi (W) \). Let \(y^{'}\in V \) to show that \(y^{'}\in \phi (W)\), i.e., show that there exists a L-fuzzy path in W whose end point is \(y^{'}\). Define a L-fuzzy path \(\omega ^{'}\) from \( \omega (t^{'})\) to \(y^{'}\). Now define a L-fuzzy path \(\bar{\omega }: I\rightarrow Y\)

$$\begin{aligned} \bar{\omega }(t)={\left\{ \begin{array}{ll} \omega (t), &{} 0\le t \le t' \\ \omega '\left( \frac{t-t'}{1-t'}\right) , &{} t' \le t \le 1 \end{array}\right. } \end{aligned}$$

For \(i=j+1,j+2,\cdots ,n\), \(\bar{\omega }(K_i)=\omega (K_i)\subset U_i\).

$$\begin{aligned} \therefore \bar{\omega }\in W(K_i, U_i). \end{aligned}$$

For \(i=1,2,\cdots ,j\),

$$\begin{aligned} \bar{\omega }(K_i)&=\bar{\omega }[K_i \cap [0,t']]\cup \bar{\omega }[K_i \cap [t',1]]\\&\subset \omega (K_i)\cup \omega '(I)\\&\subset U_i \cup V =U_i \end{aligned}$$

We get \(\bar{\omega }(K_i)\subset U_i\), \(i=1,2,\cdots \). Let \(\bar{\omega }\in W(K_i, U_i)\), so \(\bar{\omega }\in \cap _{i=1}^{n}W(K_i, U_i)=W\). This implies \( \bar{\omega }\in W\). Since \(\phi (\bar{\omega })=\bar{\omega }(1)=Y'\) we get \(V \subset \phi (\omega ) \Rightarrow \phi \) is L-fuzzy open.

Now we have to show that \(\phi \) is onto. Clearly \( \phi \) is L-fuzzy path connected as \( \phi \) is L-fuzzy connected and L-fuzzy locally path connected. Let \(y \in Y\), then \(\exists \;\omega \) from \(y_0\) to y imply \(\omega \in P(Y, y_0)\) and \(\phi (\omega )=\) end point of \(\omega =y_1\). This implies \( \phi \) is onto. \(\square \)

5 Conclusion and future work

In this work, the concept of L-fuzzy compact open topology has been introduced along with some basic theories. This work will lay the foundation for further research on L-fuzzy compact open topology. We hope to build a concept of fuzzy higher homotopy groups and fuzzy universal covering space using this concept. Further, this topic can be expanded in fuzzy category theory in future.