PROBLEM SET B 1) Let $$ Y=\{a, b, c, d, e\} $$. Find the topology generated on $$ Y $$ by the class $$ Z $$ of subsets of $$ Y $$ given that $$ Z=\{\{a, b, c\},\{c, d\},\{d, e\}\} $$

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To find the topology generated on the set $$ Y = \{a, b, c, d, e\} $$ by the class $$ Z = \{\{a, b, c\}, \{c, d\}, \{d, e\}\} $$, we need to determine the smallest topology that contains all the sets in $$ Z $$.

A topology on a set $$ Y $$ is a collection of subsets of $$ Y $$ that satisfies the following properties:

1. The empty set $$ \emptyset $$ and the entire set $$ Y $$ are in the topology.
2. The union of any collection of sets in the topology is also in the topology.
3. The intersection of any finite number of sets in the topology is also in the topology.

### Step 1: Include the empty set and the entire set
We start by including the empty set and the entire set $$ Y $$:
$$
\mathcal{T} = \{\emptyset, Y\}
$$

### Step 2: Include the sets in $$ Z $$
Next, we include the sets in $$ Z $$:
$$
\mathcal{T} = \{\emptyset, Y, \{a, b, c\}, \{c, d\}, \{d, e\}\}
$$

### Step 3: Find unions of sets in $$ Z $$
Now we compute the unions of the sets in $$ Z $$:
- $$ \{a, b, c\} \cup \{c, d\} = \{a, b, c, d\} $$
- $$ \{a, b, c\} \cup \{d, e\} = \{a, b, c, d, e\} = Y $$ (already included)
- $$ \{c, d\} \cup \{d, e\} = \{c, d, e\} $$
- $$ \{a, b, c\} \cup \{c, d\} \cup \{d, e\} = \{a, b, c, d, e\} = Y $$ (already included)

So, we add the new unions:
$$
\mathcal{T} = \{\emptyset, Y, \{a, b, c\}, \{c, d\}, \{d, e\}, \{a, b, c, d\}, \{c, d, e\}\}
$$

### Step 4: Find intersections of sets in $$ Z $$
Next, we compute the intersections of the sets in $$ Z $$:
- $$ \{a, b, c\} \cap \{c, d\} = \{c\} $$
- $$ \{a, b, c\} \cap \{d, e\} = \emptyset $$ (already included)
- $$ \{c, d\} \cap \{d, e\} = \{d\} $$

So, we add the new intersections:
$$
\mathcal{T} = \{\emptyset, Y, \{a, b, c\}, \{c, d\}, \{d, e\}, \{a, b, c, d\}, \{c, d, e\}, \{c\}, \{d\}\}
$$

### Step 5: Check for additional unions and intersections
Now we check if there are any additional unions or intersections that we haven't included yet:
- The union of $$ \{c\} $$ and $$ \{d\} $$ gives $$ \{c, d\} $$ (already included).
- The union of $$ \{c\} $$ with any other set does not yield new sets.
- The intersection of any combination of the sets does not yield new sets.

### Final topology
Thus, the topology generated by the class $$ Z $$ is:
$$
\mathcal{T} = \{\emptyset, Y, \{a, b, c\}, \{c, d\}, \{d, e\}, \{a, b, c, d\}, \{c, d, e\}, \{c\}, \{d\}\}
$$

This is the final topology generated by the class $$ Z $$ on the set $$ Y $$.
The topology generated by $$ Z $$ on $$ Y $$ is $$ \{\emptyset, Y, \{a, b, c\}, \{c, d\}, \{d, e\}, \{a, b, c, d\}, \{c, d, e\}, \{c\}, \{d\}\} $$.

PROBLEM SET B

Let . Find the topology generated on by the class of subsets of
given that

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PROBLEM SET B Let . Find the topology generated on by the class of subsets of given that