Continuous Functions in Ideal Bitopological Spaces
... Definition 3.1: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)-I - continuous if f-1(V) is (i,j)-I - open in X for every σi -open set V in Y; i, j=1, 2, i ≠ j Definition 3.2: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)- precontinuous if f-1(V) is (i,j)- preopen in ...
... Definition 3.1: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)-I - continuous if f-1(V) is (i,j)-I - open in X for every σi -open set V in Y; i, j=1, 2, i ≠ j Definition 3.2: A function f: (X, τ1, τ2, I) (Y, σ1, σ2) is said to be (i,j)- precontinuous if f-1(V) is (i,j)- preopen in ...
Set representations of abstract lattices
... all closed sets of X. (C(X), ⊆ ) is a set lattice. If a lattice L is isomorphic to (C(X), ⊆ ) for a topological space X, then X is called a topological representation of L. A lattice that has a topological representation is called a C- lattices. ...
... all closed sets of X. (C(X), ⊆ ) is a set lattice. If a lattice L is isomorphic to (C(X), ⊆ ) for a topological space X, then X is called a topological representation of L. A lattice that has a topological representation is called a C- lattices. ...
On Noetherian Spaces - LSV
... least upper bound in U meets U . The latter crops up in domain theory, where a cpo is a partially ordered set where every directed family has a least upper bound. A topological space is Alexandroff-discrete iff every intersection of opens is again open. Equivalently, iff its topology is the Alexandr ...
... least upper bound in U meets U . The latter crops up in domain theory, where a cpo is a partially ordered set where every directed family has a least upper bound. A topological space is Alexandroff-discrete iff every intersection of opens is again open. Equivalently, iff its topology is the Alexandr ...
Point Set Topology
... Definition 1.25 Let X be a topological space. (1) X is T0 if for each pair of distinct points, at least one has a neighbourhood not containing the other. (2) X is T1 or Fréchet if for any distinct x, y ∈ X, there exist open neighbourhoods U of x and V of y such that y is not in U and x is not in V ...
... Definition 1.25 Let X be a topological space. (1) X is T0 if for each pair of distinct points, at least one has a neighbourhood not containing the other. (2) X is T1 or Fréchet if for any distinct x, y ∈ X, there exist open neighbourhoods U of x and V of y such that y is not in U and x is not in V ...
PDF
... nearness relation on X if it satisfies the following conditions: for A, B ∈ P (X), 1. if A ∩ B 6= ∅, then AδB; 2. if AδB, then A 6= ∅ and B 6= ∅; 3. (symmetry) if AδB, then BδA; 4. (A1 ∪ A2 )δB iff A1 δB or A2 δB; 5. Aδ 0 B implies the existence of C ⊆ X with Aδ 0 C and (X − C)δ 0 B, where Aδ 0 B me ...
... nearness relation on X if it satisfies the following conditions: for A, B ∈ P (X), 1. if A ∩ B 6= ∅, then AδB; 2. if AδB, then A 6= ∅ and B 6= ∅; 3. (symmetry) if AδB, then BδA; 4. (A1 ∪ A2 )δB iff A1 δB or A2 δB; 5. Aδ 0 B implies the existence of C ⊆ X with Aδ 0 C and (X − C)δ 0 B, where Aδ 0 B me ...
The Lattice of Domains of an Extremally Disconnected Space 1
... that for a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too (see e.g., [15], [3]). In [23] it is proved that all domains of a given topological space form a complemented lattice. One may ask whether the lattice of a ...
... that for a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too (see e.g., [15], [3]). In [23] it is proved that all domains of a given topological space form a complemented lattice. One may ask whether the lattice of a ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.