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Topology (Defintion (X be a set and let τ be a family of subsets of X.…
Topology
Defintion
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If τ is a topology on X, then the pair (X, τ) is called a topological space.
Basis
- Let (X, τ) be topological space. A collection B of open sets is said to be a base for the τ provided that each open set is a union of sets in B. The sets in B are called basic open sets.
- Every topology is base for itself.
Example: The collection of all open intervals is a base for the usual topology on R.
- The collection of all half-open intervals of the from [a,b) is a base for the half-open interval topology on R.
- The collection of all intervals of the from [a,∞) is a base for the half-open interval topology on R.
- THEOREM: Let (X, τ) be a topology space. A collection B of open sets is a base for τ iff for each open set U and each x∈U there is a set b∈B for which x∈b⊆U.
- THEOREM: Let X be a set. A collection B of sub sets of X is a base for a topology on X iff
(a) X=⋃{b:b∈B}
(b) for any b,c ∈B and any x∈ b⋂c, there exists
d ∈B for which x∈d⊆ b⋂c.
Boundry
: Let (X,τ) be a topological space and A⊆X. A point x∈X is said to be a Boundary Point of A if x is in the closure of A but not in the interior of A
Exterior
the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S.
interior
if (X,τ) is a topological space and A⊆X then a point a∈A is called an interior point of A if there exists an open set U∈τ such that: a∈U⊆A
Closure
By U being the "smallest" closed set containing A we mean that if U and V are both closed sets containing A then A⊆U⊆V. Equivalently, the closure of A can be defined to be the the intersection of all closed sets which contain A as a subset
open set
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A subset V of R is open,iff V is equal to a union of open intervals.
A subset U of R is called an open set if U=∅ or if ∀x∈U, ∃I open interval such that x∈I⊆U
Open Half-line C={(a,∞):a∈R}∪{R,∅}
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Half-Open interval H={U⊆R:U=∅ or ∀x∈U,∃[a,b) such that,x∈[a,b)⊆U}
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