Abstract:

 

In this work we introduce and study a new class of open sets by means semi‐open sets [16] and preclosed sets [8] called Sp‐open sets. By the above mentioned sets, several new concepts such as Sp‐continuous functions, almost and weakly Sp‐continuous functions, weakly Sp‐open and weakly Sp‐ closed functions and Sp‐compactness are defined and studied.

In the light of this work, some of our main results can be listed as follows:

1. The family of Sp‐open sets forms a supratopology on a space

2. If B is clopen subset of a space X and A is Sp‐open in X, then A B is Sp‐open set in X.

3. {SO(X), RC(X)} SpO(X) SO(X).

4. The  following    statements    are    equivalent    for    the    function

f: (X, τ)  (Y, σ):

  1. f is Sp‐continuous.
  2. The inverse image of every open set in Y is Sp‐open set in X.

iii.   The inverse image of every closed set in Y is Sp‐closed set in X. iv.    For each AX, f (Spcl(A)) clf (A).

  1. For each AX, intf(A) f (Spint(A)).
  2. For each BY, Spcl(f ˉ¹ (B)) f ˉ¹ (clB). vii. For each BY, f ˉ¹ (intB) Spint(f ˉ¹ (B)).
  3. 5. Let f: XY be a function and let {A :    } be regular closed cover of

X.  If  the  restriction  f|A  :  A  Y  is  Sp‐continuous  for  each    , then f is Sp‐continuous.

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