Definition. Let T be a topology on a set X, and let B c T. Then B is a basis for the topology T if and only if every open set in J is the union of elements of B. If B E B, we say B is a basis element or basic open set. Note that B is an element of the basis B, but a subset of the space X. Note. By definition, an empty union is the empty set, so any basis B will generate the empty set as a union of none of the elements of B. (We recommend you spend an empty amount of time thinking about the empty set.) Given a topology on some space X, how can we test whether a collection of subsets forms a basis for that topology? The next theorem gives an answer. Theorem 3.1. Let (X,T) be a topological space, and let B be a collection of subsets of X. Then B is a basis for T if and only if (1) ВСГ, аnd (2) for each set U in T and point p in U there is a set V in B such that p E V C U. Theorem 3.3. Suppose X is a set and B is a collection of subsets of X. Then B is a basis for some topology on X if and only if (1) each point of X is in some element of B, and (2) if U and V are sets in B and p is a point in U n V, there is a set W in B such that PEW C (Un V).

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Prove theorem 3.3 using the given theorems and definitions from Topology Through Inquiry.

Definition. Let T be a topology on a set X, and let B c T. Then B is a basis for the
topology T if and only if every open set in J is the union of elements of B. If B E B,
we say B is a basis element or basic open set. Note that B is an element of the basis
B, but a subset of the space X.
Note. By definition, an empty union is the empty set, so any basis B will generate
the empty set as a union of none of the elements of B. (We recommend you spend an
empty amount of time thinking about the empty set.)
Given a topology on some space X, how can we test whether a collection of subsets
forms a basis for that topology? The next theorem gives an answer.
Theorem 3.1. Let (X,T) be a topological space, and let B be a collection of subsets of X.
Then B is a basis for T if and only if
(1) ВСГ, аnd
(2) for each set U in T and point p in U there is a set V in B such that p E V C U.
Theorem 3.3. Suppose X is a set and B is a collection of subsets of X. Then B is a basis
for some topology on X if and only if
(1) each point of X is in some element of B, and
(2) if U and V are sets in B and p is a point in U n V, there is a set W in B such that
PEW C (Un V).
Transcribed Image Text:Definition. Let T be a topology on a set X, and let B c T. Then B is a basis for the topology T if and only if every open set in J is the union of elements of B. If B E B, we say B is a basis element or basic open set. Note that B is an element of the basis B, but a subset of the space X. Note. By definition, an empty union is the empty set, so any basis B will generate the empty set as a union of none of the elements of B. (We recommend you spend an empty amount of time thinking about the empty set.) Given a topology on some space X, how can we test whether a collection of subsets forms a basis for that topology? The next theorem gives an answer. Theorem 3.1. Let (X,T) be a topological space, and let B be a collection of subsets of X. Then B is a basis for T if and only if (1) ВСГ, аnd (2) for each set U in T and point p in U there is a set V in B such that p E V C U. Theorem 3.3. Suppose X is a set and B is a collection of subsets of X. Then B is a basis for some topology on X if and only if (1) each point of X is in some element of B, and (2) if U and V are sets in B and p is a point in U n V, there is a set W in B such that PEW C (Un V).
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