A subset $S$ of $(X,tau )$ is said to be:
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$ alpha$-open in$(X,tau )$ if $ S subset int(cl(int(S)))_{tau}$
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pre-open in $(X,tau )$ if $S subset int(cl(S))_{tau}$
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semi-preopen in$(X,tau )$ if $S subset cl(int(cl(S)))_{tau}$
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regular-open in$(X,tau )$ if $S = int(cl(S))_{tau}$.
The familly of all $ alpha$-open (Res. pre-open , semi-preopen, regular-open) subset of $(X,tau )$ is denoted by$ tau^{alpha} $(Res. $PO(X,tau), SPO(X,tau), RO(X,tau)$).
For every space $(X, tau )$ the family $ tau^{alpha}$ forms a topology on $X$ and $tau subset tau^{alpha}$.
Let$ Y subseteq X , Y in SPO(X,tau) $, which one is correct?
- if $V in PO(X,tau)$, then $ Y cap V in PO(Y, tau_{Y}) $.
- if $V in tau^{alpha} $, then $ Y cap V in tau^{alpha}(Y,tau_{Y})$.
- if $V in RO(X,tau)$, then $ Y cap V in RO(Y, tau_{Y}) $.
( $tau_{Y}$ is the subspace topology of $Y$.)