Introduction
Preliminaries
Approximations using somewhat open sets
\(\rho so\)-Lower and \(\rho so\)-upper approximations
tv | tw | tx | ty | |
---|---|---|---|---|
\({\mathcal {N}}_r\) | \(\{tw, tx\}\) | \(\emptyset \) | \(\{tw, tx\}\) | \(\{ty\}\) |
\({\mathcal {N}}_l\) | \(\emptyset \) | \(\{tv, tx\}\) | \(\{tv, tx\}\) | \(\{ty\}\) |
\({\mathcal {N}}_i\) | \(\emptyset \) | \(\emptyset \) | \(\{tx\}\) | \(\{ty\}\) |
\({\mathcal {N}}_u\) | \(\{tw, tx\}\) | \(\{tv, tx\}\) | \(\{tv, tw, tx\}\) | \(\{ty\}\) |
\({\mathcal {N}}_{\langle r\rangle }\) | \(\emptyset \) | \(\{tw, tx\}\) | \(\{tw, tx\}\) | \(\{ty\}\) |
\({\mathcal {N}}_{\langle l\rangle }\) | \(\{tv, tx\}\) | \(\emptyset \) | \(\{tv, tx\}\) | \(\{ty\}\) |
\({\mathcal {N}}_{\langle i\rangle }\) | \(\emptyset \) | \(\emptyset \) | \(\{tx\}\) | \(\{ty\}\) |
\({\mathcal {N}}_{\langle u\rangle }\) | \(\{tv, tx\}\) | \(\{tw, tx\}\) | \(\{tv, tw, tx\}\) | \(\{ty\}\) |
\(\varOmega \) | \(\underline{{\mathcal {E}}}^{so}_u(\varOmega )\) | \(\overline{{\mathcal {E}}}^{so}_u(\varOmega )\) | \(M^{so}_{u}(\varOmega )\) | \(\underline{{\mathcal {E}}}^{so}_r(\varOmega )\) | \(\overline{{\mathcal {E}}}^{so}_r(\varOmega )\) | \(M^{so}_{r}(\varOmega )\) | \(\underline{{\mathcal {E}}}^{so}_l(\varOmega )\) | \(\overline{{\mathcal {E}}}^{so}_l(\varOmega )\) | \(M^{so}_{l}(\varOmega )\) | \(\underline{{\mathcal {E}}}^{so}_i(\varOmega )\) | \(\overline{{\mathcal {E}}}^{so}_i\) | \(M^{so}_{i}(\varOmega )\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\{tv\}\) | \(\emptyset \) | \(\{tv\}\) | 0 | \(\emptyset \) | \(\{tv\}\) | 0 | \(\{tv\}\) | \(\{tv\}\) | 1 | \(\{tv\}\) | \(\{tv\}\) | 1 |
\(\{tw\}\) | \(\emptyset \) | \(\{tw\}\) | 0 | \(\{tw\}\) | \(\{tw\}\) | 1 | \(\emptyset \) | \(\{tw\}\) | 0 | \(\{tw\}\) | \(\{tw\}\) | 1 |
\(\{tx\}\) | \(\emptyset \) | \(\{tx\}\) | 0 | \(\emptyset \) | \(\{tx\}\) | 0 | \(\emptyset \) | \(\{tx\}\) | 0 | \(\{tx\}\) | \(\{tx\}\) | 1 |
\(\{ty\}\) | \(\{ty\}\) | \(\{ty\}\) | 1 | \(\{ty\}\) | \(\{ty\}\) | 1 | \(\{ty\}\) | \(\{ty\}\) | 1 | \(\{ty\}\) | \(\{ty\}\) | 1 |
\(\{tv,tw\}\) | \(\emptyset \) | \(\{tv,tw\}\) | 0 | \(\{tv,tw\}\) | \(\{tv,tw\}\) | 1 | \(\{tv,tw\}\) | \(\{tv,tw\}\) | 1 | \(\{tv,tw\}\) | \(\{tv,tw\}\) | 1 |
\(\{tv,tx\}\) | \(\emptyset \) | \(\{tv,tx\}\) | 0 | \(\emptyset \) | \(\{tv,tx\}\) | 0 | \(\{tv,tx\}\) | \(\{tv,tx\}\) | 1 | \(\{tv,tx\}\) | \(\{tv,tx\}\) | 1 |
\(\{tv,ty\}\) | \(\{tv,ty\}\) | U | \(\frac{1}{2}\) | \(\{tv,ty\}\) | \(\{tv,ty\}\) | 1 | \(\{tv,ty\}\) | U | \(\frac{1}{2}\) | \(\{tv,ty\}\) | \(\{tv,ty\}\) | 1 |
\(\{tw,tx\}\) | \(\emptyset \) | \(\{tw,tx\}\) | 0 | \(\{tw,tx\}\) | \(\{tw,tx\}\) | 1 | \(\emptyset \) | \(\{tw,tx\}\) | 0 | \(\{tw,tx\}\) | \(\{tw,tx\}\) | 1 |
\(\{tw,ty\}\) | \(\{tw,ty\}\) | U | \(\frac{1}{2}\) | \(\{tw,ty\}\) | U | \(\frac{1}{2}\) | \(\{tw,ty\}\) | \(\{tw,ty\}\) | 1 | \(\{tw,ty\}\) | \(\{tw,ty\}\) | 1 |
\(\{tx,ty\}\) | \(\{tx,ty\}\) | U | \(\frac{1}{2}\) | \(\{tx,ty\}\) | \(\{tx,ty\}\) | 1 | \(\{tx,ty\}\) | \(\{tx,ty\}\) | 1 | \(\{tx,ty\}\) | \(\{tx,ty\}\) | 1 |
\(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | 1 | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | 1 | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | 1 | \(\{tv,tw,tx\}\) | \(\{tv,tw, tx\}\) | 1 |
\(\{tv,tw,ty\}\) | \(\{tv,tw,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tw,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tw,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tw,ty\}\) | \(\{tv,tw,ty\}\) | 1 |
\(\{tv,tx,ty\}\) | \(\{tv,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tx,ty\}\) | \(\{tv,tx,ty\}\) | 1 | \(\{tv,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tx,ty\}\) | \(\{tv,tx,ty\}\) | 1 |
\(\{tw,tx,ty\}\) | \(\{tw,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tw,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tw,tx,ty\}\) | \(\{tw,tx,ty\}\) | 1 | \(\{tw,tx,ty\}\) | \(\{tw,tx,ty\}\) | 1 |
U | U | U | 1 | U | U | 1 | U | U | 1 | U | U | 1 |
\(\varOmega \) | \(\underline{{\mathcal {E}}}^{so}_{\langle u\rangle }(\varOmega )\) | \(\overline{{\mathcal {E}}}^{so}_{\langle u\rangle }(\varOmega )\) | \(M^{so}_{\langle u\rangle }(\varOmega )\) | \(\underline{{\mathcal {E}}}^{so}_{\langle r\rangle }(\varOmega )\) | \(\overline{{\mathcal {E}}}^{so}_{\langle r\rangle }(\varOmega )\) | \(M^{so}_{\langle r\rangle }(\varOmega )\) | \(\underline{{\mathcal {E}}}^{so}_{\langle l\rangle }(\varOmega )\) | \(\overline{{\mathcal {E}}}^{so}_{\langle l\rangle }(\varOmega )\) | \(M^{so}_{\langle l\rangle }(\varOmega )\) | \(\underline{{\mathcal {E}}}^{so}_{\langle i\rangle }(M)\) | \(\overline{{\mathcal {E}}}^{so}_{\langle i\rangle }\) | \(M^{so}_{\langle i\rangle }(M)\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\{tv\}\) | \(\emptyset \) | \(\{tv\}\) | 0 | \(\{tv\}\) | \(\{tv\}\) | 1 | \(\emptyset \) | \(\{tv\}\) | 0 | \(\{tv\}\) | \(\{tv\}\) | 1 |
\(\{tw\}\) | \(\emptyset \) | \(\{tw\}\) | 0 | \(\emptyset \) | \(\{tw\}\) | 0 | \(\{tw\}\) | \(\{tw\}\) | 1 | \(\{tw\}\) | \(\{tw\}\) | 1 |
\(\{tx\}\) | \(\emptyset \) | \(\{tx\}\) | 0 | \(\emptyset \) | \(\{tx\}\) | 0 | \(\emptyset \) | \(\{tx\}\) | 0 | \(\{tx\}\) | \(\{tx\}\) | 1 |
\(\{ty\}\) | \(\{ty\}\) | \(\{ty\}\) | 1 | \(\{ty\}\) | \(\{ty\}\) | 1 | \(\{ty\}\) | \(\{ty\}\) | 1 | \(\{ty\}\) | \(\{ty\}\) | 1 |
\(\{tv,tw\}\) | \(\emptyset \) | \(\{tv,tw\}\) | 0 | \(\{tv,tw\}\) | \(\{tv,tw\}\) | 1 | \(\{tv,tw\}\) | \(\{tv,tw\}\) | 1 | \(\{tv,tw\}\) | \(\{tv,tw\}\) | 1 |
\(\{tv,tx\}\) | \(\emptyset \) | \(\{tv,tx\}\) | 0 | \(\{tv,tx\}\) | \(\{tv,tx\}\) | 1 | \(\{tv,tx\}\) | \(\{tv,tx\}\) | 1 | \(\{tv,tx\}\) | \(\{tv,tx\}\) | 1 |
\(\{tv,ty\}\) | \(\{tv,ty\}\) | U | \(\frac{1}{2}\) | \(\{tv,ty\}\) | \(\{tv,ty\}\) | 1 | \(\{tv,ty\}\) | \(\{tv,ty\}\) | 1 | \(\{tv,ty\}\) | \(\{tv,ty\}\) | 1 |
\(\{tw,tx\}\) | \(\emptyset \) | \(\{tw,tx\}\) | 0 | \(\{tw,tx\}\) | \(\{tw,tx\}\) | 1 | \(\{tw,tx\}\) | \(\{tw,tx\}\) | 1 | \(\{tw,tx\}\) | \(\{tw,tx\}\) | 1 |
\(\{tw,ty\}\) | \(\{tw,ty\}\) | U | \(\frac{1}{2}\) | \(\{tw,ty\}\) | \(\{tw,ty\}\) | 1 | \(\{tw,ty\}\) | \(\{tw,ty\}\) | 1 | \(\{tw,ty\}\) | \(\{tw,ty\}\) | 1 |
\(\{tx,ty\}\) | \(\{tx,ty\}\) | U | \(\frac{1}{2}\) | \(\{tx,ty\}\) | \(\{tx,ty\}\) | 1 | \(\{tx,ty\}\) | \(\{tx,ty\}\) | 1 | \(\{tx,ty\}\) | \(\{tx,ty\}\) | 1 |
\(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | 1 | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | 1 | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | 1 | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | 1 |
\(\{tv,tw,ty\}\) | \(\{tv,tw,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tw,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tw,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tw,ty\}\) | \(\{tv,tw,ty\}\) | 1 |
\(\{tv,tx,ty\}\) | \(\{tv,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tv,tx,ty\}\) | \(\{tv,tx,ty\}\) | 1 | \(\{tv,tx,ty\}\) | \(\{tv,tx,ty\}\) | 1 |
\(\{tw,tx,ty\}\) | \(\{tw,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tw,tx,ty\}\) | \(\{tw,tx,ty\}\) | 1 | \(\{tw,tx,ty\}\) | U | \(\frac{3}{4}\) | \(\{tw,tx,ty\}\) | \(\{tw,tx,ty\}\) | 1 |
U | U | U | 1 | U | U | 1 | U | U | 1 | U | U | 1 |
Comparison of our approach with the previous ones
\(\vartheta _\rho \) | ||||||
---|---|---|---|---|---|---|
P(U) | \(\vartheta _r\) | \(semiO(\vartheta _r)\) | \(so(\vartheta _r)\) | |||
\(\underline{{\mathcal {E}}}_r\) | \(\overline{{\mathcal {E}}}_r\) | \(\underline{{\mathcal {E}}}^{semi}_r\) | \(\overline{{\mathcal {E}}}^{semi}_r\) | \(\underline{{\mathcal {E}}}^{so}_r\) | \(\overline{{\mathcal {E}}}^{so}_r\) | |
\(\{tv\}\) | \(\emptyset \) | \(\{tv\}\) | \(\emptyset \) | \(\{tv\}\) | \(\emptyset \) | \(\{tv\}\) |
\(\{{tw}\}\) | \(\{{tw}\}\) | \(\{{tv,tx,ty}\}\) | \(\{{tw}\}\) | \(\{{tw,tx}\}\) | \(\{{tw}\}\) | \(\{{tw}\}\) |
\(\{tx\}\) | \(\emptyset \) | \(\{tv,tx\}\) | \(\emptyset \) | \(\{tx\}\) | \(\emptyset \) | \(\{tx\}\) |
\(\{ty\}\) | \(\{ty\}\) | \(\{ty\}\) | \(\{ty\}\) | \(\{ty\}\) | \(\{ty\}\) | \(\{ty\}\) |
\(\{{tv,tw}\}\) | \(\{{tw}\}\) | \(\{{tv,tw,tx}\}\) | \(\{{tv,tw}\}\) | \(\{{tv,tw,tx}\}\) | \(\{{tv,tw}\}\) | \(\{{tv,tw}\}\) |
\(\{tv,tx\}\) | \(\emptyset \) | \(\{tv,tx\}\) | \(\emptyset \) | \(\{tv,tx\}\) | \(\emptyset \) | \(\{tv,tx\}\) |
\(\{tv,ty\}\) | \(\{ty\}\) | \(\{tv,ty\}\) | \(\{tv,ty\}\) | \(\{tv,ty\}\) | \(\{tv,ty\}\) | \(\{tv,ty\}\) |
\(\{tw,tx\}\) | \(\{tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tw,tx\}\) | \(\{tw,tx\}\) |
\(\{tw,ty\}\) | \(\{tw,ty\}\) | U | \(\{tw,ty\}\) | U | \(\{tw,ty\}\) | U |
\(\{{tx,ty}\}\) | \(\{{ty}\}\) | \(\{{tv,tx,ty}\}\) | \(\{{ty}\}\) | \(\{{tx,ty}\}\) | \(\{{tx,ty}\}\) | \(\{{tx,ty}\}\) |
\(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) | \(\{tv,tw,tx\}\) |
\(\{tv,tw,ty\}\) | \(\{tw,ty\}\) | U | \(\{tv,tw,ty\}\) | U | \(\{tv,tw,ty\}\) | U |
\(\{{tv,tx,ty}\}\) | \(\{{ty}\}\) | \(\{{tv,tx,ty}\}\) | \(\{{tv,ty}\}\) | \(\{{tv,tx,ty}\}\) | \(\{{tv,tx,ty}\}\) | \(\{{tv,tx,ty}\}\) |
\(\{tw,tx,ty\}\) | \(\{tw,tx,ty\}\) | U | \(\{tw,tx,ty\}\) | U | \(\{tw,tx,ty\}\) | U |
Accuracy | |||
---|---|---|---|
P(U) | \(M_r\) | \(M^{semi}_r\) | \(M^{so}_r\) |
\(\{tv\}\) | 0 | 0 | 0 |
\(\{{tw}\}\) | \(\frac{1}{3}\) | \(\frac{1}{2}\) | 1 |
\(\{tx\}\) | 0 | 0 | 0 |
\(\{ty\}\) | 1 | 1 | 1 |
\(\{{tv,tw}\}\) | \({\frac{1}{3}}\) | \({\frac{2}{3}}\) | 1 |
\(\{tv,tx\}\) | 0 | 0 | 0 |
\(\{tv,ty\}\) | \(\frac{1}{2}\) | 1 | 1 |
\(\{tw,tx\}\) | \(\frac{2}{3}\) | \(\frac{2}{3}\) | 1 |
\(\{tw,ty\}\) | \(\frac{1}{3}\) | \(\frac{1}{2}\) | \(\frac{1}{2}\) |
\(\{{tx,ty}\}\) | \({\frac{1}{3}}\) | \({\frac{1}{2}}\) | 1 |
\(\{tv,tw,tx\}\) | 1 | 1 | 1 |
\(\{tv,tw,ty\}\) | \(\frac{1}{2}\) | \(\frac{3}{4}\) | \(\frac{3}{4}\) |
\(\{{tv,tx,ty}\}\) | \({\frac{1}{3}}\) | \({\frac{2}{3}}\) | 1 |
\(\{tw,tx,ty\}\) | \(\frac{3}{4}\) | \(\frac{3}{4}\) | \(\frac{3}{4}\) |
Medical example: Dengue fever
U | J | H | S | T | Dengue fever |
---|---|---|---|---|---|
\(\mu _{1}\) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | h | \(\checkmark \) |
\(\mu _{2}\) | \(\checkmark \) | \(\times \) | \(\times \) | h | \(\times \) |
\(\mu _{3}\) | \(\checkmark \) | \(\times \) | \(\times \) | h | \(\checkmark \) |
\(\mu _{4} \) | \(\times \) | \(\times \) | \(\times \) | vh | \(\times \) |
\(\mu _{5}\) | \(\times \) | \(\checkmark \) | \(\checkmark \) | h | \(\times \) |
\(\mu _{6}\) | \(\checkmark \) | \(\checkmark \) | \(\times \) | vh | \(\checkmark \) |
\(\mu _{7}\) | \(\checkmark \) | \(\checkmark \) | \(\times \) | n | \(\times \) |
\(\mu _{8}\) | \(\checkmark \) | \(\checkmark \) | \(\times \) | vh | \(\checkmark \) |
\(\mu _{1}\) | \(\mu _{2}\) | \(\mu _{3}\) | \(\mu _{4} \) | \(\mu _{5}\) | \(\mu _{6}\) | \(\mu _{7}\) | \(\mu _{8}\) | |
---|---|---|---|---|---|---|---|---|
\(\mu _{1}\) | 1 | 0.5 | 0.5 | 0 | 0.75 | 0.5 | 0.5 | 0.5 |
\(\mu _{2}\) | 0.5 | 1 | 1 | 0.5 | 0.25 | 0.5 | 0.5 | 0.5 |
\(\mu _{3}\) | 0.5 | 1 | 1 | 0.5 | 0.25 | 0.5 | 0.5 | 0.5 |
\(\mu _{4} \) | 0 | 0.5 | 0.5 | 1 | 0.25 | 0.5 | 0.25 | 0.5 |
\(\mu _{5}\) | 0.75 | 0.25 | 0.25 | 0.25 | 1 | 0.25 | 0.25 | 0.25 |
\(\mu _{6}\) | 0.5 | 0.5 | 0.5 | 0.5 | 0.25 | 1 | 0.75 | 1 |
\(\mu _{7}\) | 0.5 | 0.5 | 0.5 | 0.25 | 0.25 | 0.75 | 1 | 0.75 |
\(\mu _{8}\) | 0.5 | 0.5 | 0.5 | 0.5 | 0.25 | 1 | 0.75 | 1 |
\({\mathcal {N}}_\rho \) | |
---|---|
\(\mu _1\) | \(\{\mu _1, \mu _5\}\) |
\(\mu _2\) | \(\{\mu _2, \mu _3\}\) |
\(\mu _3\) | \(\{\mu _2, \mu _3\}\) |
\(\mu _4\) | \(\{\mu _4\}\) |
\(\mu _5\) | \(\{\mu _1, \mu _5\}\) |
\(\mu _6\) | \(\{\mu _6, \mu _7, \mu _8\}\) |
\(\mu _7\) | \(\{\mu _6, \mu _7, \mu _8\}\) |
\(\mu _8\) | \(\{\mu _6, \mu _7, \mu _8\}\) |
Discussion: strengths and limitations
-
Strengths1.Our approach preserves the monotonic property for the accuracy and roughness measures (see, Proposition 6 and Corollary 2); whereas, this property is losing in the previous topological approaches given in [14, 37]. This is due to that our approach is only based on the interior operator which is proportional to the size of a given topology. However, the other approaches are based on two factors, interior and closure operators, which are working against each other with respect to the size of a given topology. That is, when the size of a given topology enlarges, the interior points of a subset is increasing and the closure points of a subset are decreasing which means that we cannot anticipate the behaviours of the approximations in cases of \(\alpha \)-open, semi-open, pre-open, b-open, and \(\beta \)-open and somewhere dense sets.2.All Pawlak properties are preserved by \(\rho so\)-lower and \(\rho so\)-upper approximations except for (L5) and (U6) given in Proposition 1 (see their counterparts:(v) and (vi) given, respectively, in Proposition 3 and Proposition 4). These two properties are kept by \(\rho so\)-approximations under a hyperconnectedness condition, whereas we need a strong hyperconnectedness condition to keep them by the approximations generated from somewhere dense sets. That is, the properties (L5) and (U6) are preserved by \(\rho so\)-approximations under relaxed conditions than the other approximations.3.Comparisons between the different types of \(\rho so\)-approximations and \(\rho so\)-accuracy measures are investigated in Proposition 10 and Corollary 4. Whereas, we cannot compare between the different types of approximations and accuracy measures induced from \(\alpha \)-open and \(\alpha \)-closed sets, because they are defined using interior and closure operators which are working against each other. This matter does not guarantee standard behaviour between \(\rho \alpha \)-approximations and \(\rho \alpha \)-accuracy measures. For the same reason, this matter applied to the other approximations and accuracy measures induced from semi-open, pre-open, b-open, \(\beta \)-open sets, and somewhere dense sets.
-
limitations1.Our approach is incomparable with those given in [14, 37] in cases of pre-open, b-open, and \(\beta \)-open sets. To validate this matter, consider the collections given in (3), and let \(\varOmega =\{tx,ty\}\) and \(\Sigma =\{tv\}\) be subsets of \((U, \vartheta _{r})\) and \((U, \vartheta _{u})\), respectively. By calculation, we find that \(cl(int(cl(\varOmega )))=\{ty\}\) and \(int(\varOmega )=\{ty\}\) which means that \(\varOmega \) is somewhat open, but not pre-open (b-open, \(\beta \)-open). Also, \(int(cl(\Sigma ))=\{tv,tw,tx\}\) and \(int(\Sigma )=\emptyset \) which means that \(\Sigma \) is pre-open (b-open, \(\beta \)-open), but not somewhat open. However, the accuracy measures and approximations generated by the class of pre-open subsets are better than our approach under a finite quasi-discrete topology, because all subsets of a finite quasi-discrete topology are pre-open; hence, the accuracy measures induced from this class are equal to one for any subset; this matter is also applied to all classes that are wider than the class of pre-open sets such as b-open, \(\beta \)-open, and somewhere dense sets.2.One can note that every somewhat open set is somewhere dense; so that, \(\underline{{\mathcal {E}}}^{so}_\rho (\varOmega )\subseteq \underline{{\mathcal {E}}}^{SD}_\rho (\varOmega )\subseteq \varOmega \subseteq \overline{{\mathcal {E}}}^{SD}_\rho (\varOmega )\subseteq \overline{{\mathcal {E}}}^{so}_\rho (\varOmega )\). Consequently, \(M^{so}_\rho (\varOmega )\le M^{SD}_\rho (\varOmega )\). Hence, the approximations and accuracy measures generated from the method of somewhere dense sets given in [8] are better than their counterparts given in this manuscript.