Please solve THEOREM 2.47 in DETAIL. i posted this earlier and someone posted the wrong answer from CHEGG, so please dont do that. only attempt if you know the answer. Definition 2.44. A set S is a subbasis for a topology on a set X provided US = X.Х.The topology T on X generated by S is defined to be the collection of all unions of finiteintersections of elements of S.Problem 2.45. Can you define the basis generated by a subbasis?Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) Thesets in the subbasis are open sets in that topology.Theorem 2.47. Suppose B is a basis for a topology on X and Y C X. Then By := {BnYBE B} is a basis for the subspace topology on Y.

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Description: Please solve THEOREM 2.47 in DETAIL. i posted this earlier and someone posted the wrong answer from CHEGG, so please dont do that. only attempt if you know the answer. Definition 2.44. A set S is a subbasis for a topology on a set X provided US = X.Х

Answered: Theorem 2.47. Suppose B is a basis for…

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Please solve THEOREM 2.47 in DETAIL. i posted this earlier and someone posted the wrong answer from CHEGG, so please dont do that. only attempt if you know the answer.

Definition 2.44. A set S is a subbasis for a topology on a set X provided US = X.
Х.
The topology T on X generated by S is defined to be the collection of all unions of finite
intersections of elements of S.
Problem 2.45. Can you define the basis generated by a subbasis?
Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The
sets in the subbasis are open sets in that topology.
Theorem 2.47. Suppose B is a basis for a topology on X and Y C X. Then By := {BnY
BE B} is a basis for the subspace topology on Y.

Expert Solution

Step 1

Definition: Basis for a topology

Let $(X, τ)$ be a topological space and $Y$ be a subset of $X$ . A collection of open sets $B$ is said to be a basis for the topology $τ$ , if it satisfies the following conditions.

For every element $x$ in $X$ , there is a set $B$ in the collection of sets such that $x \in B$ .
Suppose $x \in B_{1} \cap B_{2}$ , where $B_{1}$ and $B_{2}$ in the collection $B$ , then there exists a set $B_{3}$ in $B$ such that $x \in B_{3}$ and $B_{3} \subseteq B_{1} \cap B_{2}$ .

Definition: Subspace topology

Let $(X, τ)$ be a topological space and $B$ be a basis for the topology. Consider a subset $Y$ of $x$ . The subspace topology on $Y$ is defined as follows,

$τ_{Y} = \{Y \cap U : U \in τ\}$

The subspace topology obtained by intersecting all the sets in $τ$ with $Y$ .

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