Random Texts - 28 – 2 February 2014 Definition 1.5 (a) An - TopicsExpress

Random Texts - 28 – 2 February 2014 Definition 1.5 (a) An isolated point of a set A is a point such that on some small limb of this point it is the only point of A inside this limb. That is, it is a point such that its intersection with some small limb is the point itself. (b) A boundary point of a set A is a point which is on the boundary of the set A. That is, considering any limb of this point we can find points from the set A and from outside the set A. As an example we can consider the set A = [1,2]⋃{3}. We have that 3 is an isolated point, and 1,2 and 3 are boundary points. In Performance Art field we consider a moment in time, space and movement as an isolated point, depending of what set we are considering, such that in its limb it is the only point belonging to the set that is in the limb. If for instance the performer is silent for 30 seconds and then screams, this moment of screaming can be considered an isolated point regarding the silent/screaming environment. It is also a boundary point. Let us now finally introduce the points in which we construct limbs. In order to do this, we need first to define real valued mapping or function, which can be seen as a generalization of a sequence. Definition 1.6 A function f is an operation which maps elements of C ⊆ℝ into D⊆ℝ, that is C → D x ↦ f(x) For each element x∈ C we correspond one and only one element y= f(x)∈ D. We usually call C the domain of the function f, i.e., C= dom f. Now we define the points where we will consider neighborhoods Definition 1.7 A cut is the moment when the function takes some concrete value. This value is chosen such that it preserves the properties of the function, in the sense that it isn’t an isolated or a boundary point. Considering the same set as before A=[1,2]⋃{3}, we can choose any point from the set ]1,2[ as a candidate to be a cut. In Performance Art field and considering also the previous example, any interior moment when is performer is silent can be a cut.

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