Given X = a, b, c and τ = ∅, a, b, c, a, b, a, c, b, c, a, b, c, determine if τ is a topology on X.

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Here's the solution for each problem. Problem 1: Given X = a, b, c and τ = , a, b, c, a, b, a, c, b, c, a, b, c. We need to determine if τ is a topology on X by checking the three axioms of a topology. Step 1: Check if ∈ τ and X ∈ τ. We see that ∈ τ and X = a, b, c ∈ τ. Axiom 1 is satisfied. Step 2: Check if the intersection of any two sets in τ is also in τ. Let A, B ∈ τ. We need to verify if A ∩ B ∈ τ. Since τ contains all possible subsets of X, the intersection of any two subsets of X will also be a subset of X. Therefore, their intersection will be in τ. For example: a, b ∩ b, c = b ∈ τ; a ∩ c = ∈ τ Axiom 2 is satisfied. Step 3: Check if the union of any collection of sets in τ is also in τ. Let A_i_i ∈ I be any collection of sets in τ. We need to verify if _(i ∈ I) A_i ∈ τ. Since τ contains all possible subsets of X, the union of any collection of subsets of X will also be a subset of X. Therefore, their union will be in τ. For example: a ∪ b = a, b ∈ τ; a, b ∪ c = a, b, c = X ∈ τ Axiom 3 is satisfied. Since all three axioms are satisfied, τ is a topology on X. Problem 2: Given X = a, b, c, d and τ = , X, a, b, a, c, a, b, c. We need to determine if τ is a topology on X by checking the three axioms of a topology. Step 1: Check if ∈ τ and X ∈ τ. We see that ∈ τ and X = a, b, c, d ∈ τ. Axiom 1 is satisfied. Step 2: Check if the intersection of any two sets in τ is also in τ. Let's consider the intersection of a, b ∈ τ and a, c ∈ τ. a, b ∩ a, c = a However, a is not an element of τ. Therefore, Axiom 2 is not satisfied. Since Axiom 2 is not satisfied, τ is not a topology on X. Final Answer: 1. τ is a topology on X. 2. τ is not a topology on X.

Given X = a, b, c and τ = ∅, a, b, c, a, b, a, c, b, c, a, b, c, determine if τ is a topology on X.

Step-by-step solution

Check if $\emptyset \in \tau$ and $X \in \tau$ .

Check if the intersection of any two sets in $\tau$ is also in $\tau$ .

Check if the union of any collection of sets in $\tau$ is also in $\tau$ .

Check if $\emptyset \in \tau$ and $X \in \tau$ .

Check if the intersection of any two sets in $\tau$ is also in $\tau$ .

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More Mathematics Questions

Given X = a, b, c and τ = ∅, a, b, c, a, b, a, c, b, c, a, b, c, determine if τ is a topology on X.

Step-by-step solution

Check if ∅∈τ\emptyset \in \tau∅∈τ and X∈τX \in \tauX∈τ.

Check if the intersection of any two sets in τ\tauτ is also in τ\tauτ.

Check if the union of any collection of sets in τ\tauτ is also in τ\tauτ.

Check if ∅∈τ\emptyset \in \tau∅∈τ and X∈τX \in \tauX∈τ.

Check if the intersection of any two sets in τ\tauτ is also in τ\tauτ.

Still stuck on this one?

More Mathematics Questions

Check if $\emptyset \in \tau$ and $X \in \tau$ .

Check if the intersection of any two sets in $\tau$ is also in $\tau$ .

Check if the union of any collection of sets in $\tau$ is also in $\tau$ .

Check if $\emptyset \in \tau$ and $X \in \tau$ .

Check if the intersection of any two sets in $\tau$ is also in $\tau$ .