subclass_of subbase
In ___MASK33___ , ___MASK16___ ( from the Greek  , " place ", and  , " ___MASK62___ ") is concerned with the properties of space that are preserved under continuous deformations , such as stretching and bending , but not tearing or gluing . This can be studied by considering a collection of subsets , called open sets , that satisfy certain properties , turning the given set into what is known as a topological space . Important topological properties include connectedness and compactness . In ___MASK16___ , a subbase ( or subbasis ) for a ___MASK22___ X with ___MASK16___ T is a subcollection B of T that generates T , in the sense that T is the smallest ___MASK16___ containing B . A slightly different definition is used by some authors , and there are other useful equivalent formulations of the definition ; these are discussed below . ___MASK33___ ( from ___MASK52___  " máthma ",  knowledge , ___MASK62___ , learning ) is the ___MASK62___ of topics such as quantity ( numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . In ___MASK33___ , connectedness is used to refer to various properties meaning , in some ___MASK21___ , " all one piece ". When a ___MASK48___ has such a property , we say it is connected ; otherwise it is disconnected . When a disconnected ___MASK39___ can be split naturally into connected pieces , each piece is usually called a " component " ( or " connected component "). In ___MASK16___ , an open set is an abstract concept generalizing the idea of an open ___MASK38___ in the real line . The simplest example is in metric spaces , where open sets can be defined as those sets which contain an open ball around each of their points ( or , equivalently , a set is open if it doesn ' t contain any of its boundary points ); however , an open set , in general , can be very abstract : any collection of sets can be called open , as long as the union of an arbitrary number of open sets is open , the intersection of a finite number of open sets is open , and the space itself is open . These conditions are very loose , and they allow enormous flexibility in the choice of open sets . In the two extremes , every set can be open ( called the discrete ___MASK16___ ), or no set can be open but the space itself and the empty set ( the indiscrete ___MASK16___ ).