In Example 1 of [1], the authors concluded that \((X,\tau ,E)\) is a soft topological space over the universe \(X=\{h_1,h_2,h_3,h_4\}\) and the set of parameters \(E=\{e_1,e_2,e_3\}\). Actually, their conclusion is not correct where \((F_1,E)\), \((F_2,E)\in \tau \) leads to \((F_1,E){\tilde{{\cap }}}(F_2,E)\not \in \tau \) and \((F_1,E){\tilde{{\cup}}}(F_2,E)\not \in \tau \). Also, this is achieved for another soft sets like \((F_1,E)\) and \((F_{14}, E)\). As a result of this flaw, Examples 2, 6, 7 and 8 in [1] are not correct. Examples 1 and 2 achieve the goal of Examples 1 and 2 in [1].

FormalPara Example 1

Let \(X=\{h_1,h_2,h_3,h_4\}\) and \(E=\{e_1,e_2\}\).

  1. (1)

    If \(\tau _1=\{\tilde{{\emptyset}},{\tilde{{X}}},(F_1,E), (F_2,E), (F_3,E)\}\) is a soft topology where \((F_1,E)\), \((F_2,E)\), \((F_3,E)\) and (F, E) are soft sets over X defined by:

    $$\begin{aligned} \begin{array}{ll} F_1(e_1)=\{h_1\}, &{} F_1(e_2)=\{h_1\}, \\ F_2(e_1)=\{h_2\}, &{} F_2(e_2)=\{h_2\}, \\ F_3(e_1)=\{h_1,h_2\}, &{} F_3(e_2)=\{h_1,h_2\}. \\ \end{array} \end{aligned}$$

    and

    $$\begin{aligned} F(e_1)=\{h_1,h_3\},\quad F(e_2)=\{h_1,h_3\}. \end{aligned}$$

    Then (F, E) is sb-open set but not sp-open set.

  2. (2)

    If \(\tau _2=\{{\tilde{{\emptyset }}},{\tilde{{X}}},(F_1,E), \dots , (F_7,E)\}\) is a soft topology where \((F_1,E)\), …, \((F_7,E)\) and (H, E) are soft sets over X defined by:

    $$\begin{aligned} \begin{array}{ll} F_1(e_1)=\{h_1,h_2\}, &{} F_1(e_2)=\{h_1,h_2\},\\ F_2(e_1)=\{h_2\}, &{} F_2(e_2)=\{h_1,h_3\},\\ F_3(e_1)=\{h_2,h_3\}, &{} F_3(e_2)=\{h_1\},\\ F_4(e_1)=\{h_2\}, &{} F_4(e_2)=\{h_1\},\\ F_5(e_1)=\{h_1,h_2\}, &{} F_5(e_2)=X,\\ F_6(e_1)=X, &{} F_6(e_2)=\{h_1,h_2\},\\ F_7(e_1)=\{h_2,h_3\}, &{} F_7(e_2)=\{h_1,h_3\}. \end{array} \end{aligned}$$

    and

    $$\begin{aligned} H(e_1)=\emptyset ,\quad H(e_2)=\{h_1\}. \end{aligned}$$

    Then (H, E) is sb-open set but not ss-open set.

FormalPara Example 2

Let \({\mathbb {R}}\) be the set of the real numbers and \(E=\{e\}\) be the parameters set. Define the usual soft topology over \({\mathbb {R}}\) with respect to E. Let (F, E) be a soft set over \({\mathbb {R}}\) defined by \(F(e)=[0,1)\cap {\mathbb {Q}}\) where \({\mathbb {Q}}\) is the set of rational numbers, then (F, E) is s\(\beta \)-open set but not sb-open set.

The following two examples meet the purpose of Examples 6, 7 and 8 in [1].

FormalPara Example 3

Let \(X=\{h_1,h_2,h_3,h_4\}\), \(Y=\{m_1,m_2,m_3,m_4\}\), \(E=\{e_1,e_2\}\) and \(K=\{k_1,k_2\}\).

  1. 1.

    If \((X,\tau _1,E)\) is a soft topological space defined as in Example 1(1), \((Y,\nu _1,K)\) be a soft topological space defined over Y where \(\nu _1=\{{\tilde{{\emptyset}}},{\tilde{{Y}}}, (L_1,K)\}\) and \((L_1,K)\) is a soft set on Y defined by:

    $$\begin{aligned} L_1(k_1)=\{m_1,m_2\},\quad L_1(k_2)=\{m_1,m_2\}. \end{aligned}$$

    Let \(f:(X,\tau _1,E)\rightarrow (Y,\nu _1,K)\) be a soft function defined by

    $$\begin{aligned} u(h_1)=m_2,\quad u(h_2)=m_4,\quad u(h_3)=m_1,\quad u(h_4)=m_3, \end{aligned}$$
    $$\begin{aligned} p(e_1)=k_2,\quad p(e_2)=k_1. \end{aligned}$$

    Then f is soft b-continuous function but not soft pre-continuous.

  2. 2.

    If \((X,\tau _2,E)\) is a soft topological space defined as in Example 1(2), \((Y,\nu _2,K)\) is a soft topological space defined over Y where \(\nu _2=\{{\tilde{{\emptyset}}},{\tilde{{Y}}}, (L_2,K)\}\) and \((L_2,K)\) is a soft set on Y defined by:

    $$\begin{aligned} L_2(k_1)=\{m_2\},\quad L_2(k_2)=\emptyset . \end{aligned}$$

    Let \(f:(X,\tau _1,E)\rightarrow (Y,\nu _2,K)\) be a soft function defined as in (1), then f is soft b-continuous function but not soft semi-continuous.

FormalPara Example 4

Let \({\mathbb {R}}\) be the set of real numbers, \(E=\{e\}\), \(K=\{k\}\), \(({\mathbb {R}},\mathcal {U}, E)\) be the usual soft topology and \(({\mathbb {R}},\mathcal {V},K)\) be a soft topology over \({\mathbb {R}}\) such that \(\mathcal {V}=\{{\tilde{{\emptyset}}},{\tilde{{R}}}, (M,K)\}\) where (M, K) is a soft set defined over \({\mathbb {R}}\) by \(M(k)=[0,1)\cap {\mathbb {Q}}\). Define the maps \(u:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and \(p:E\rightarrow K\) by

$$\begin{aligned} u(x)=\left\{ \begin{array}{ll} x, &{} \hbox {if } x\in [0,1)\cap {\mathbb {Q}}; \\ 0, &{} \hbox {otherwise.} \end{array} \right. \end{aligned}$$

and \(p(e)=k\). Then \(f^{-1}(M,K)=(H,E)\) where \(H(e)=[0,1)\cap {\mathbb {Q}}\). Thus f is soft \(\beta \)-continuous function but not soft b-continuous.