StudyLabs

Worksheet 1

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Question 1

Let be a set and let be a collection of topologies on .

(a)

Is

a topology on ?

Answer: No, not necessarily.

A topology must be closed under arbitrary unions and finite intersections.

Although every individually satisfies these properties, their union may fail closure under finite intersections.

Counterexample

Let:

Define:

Then:

But:

Actually closure still holds here, so modify example:

Take instead:

where:

Hence the union need not be closed under intersections.

Therefore:

is not necessarily a topology.

(b) Prove

is a topology on .

Let:

We verify topology axioms.

1. Contains and

Since every is a topology:

Hence:

2. Closed under arbitrary unions

Let:

Then:

Since each is a topology:

Therefore:

3. Closed under finite intersections

If:

then:

Thus:

Hence:

Therefore:

is a topology.

(c) Prove

is the unique largest topology contained in every .

Let:

Step 1: Containment

For every :

since every element of lies in all .

Step 2: Maximality

Let be a topology such that:

Then every element of lies in all , so:

Thus is the largest topology contained in all .

Step 3: Uniqueness

If another topology had the same property:

then:

and also:

so:

Therefore the topology is unique.

(d) Prove there exists a unique smallest topology such that

Existence

Consider the family:

Define:

From part (b), is a topology.

Also:

So satisfies the condition.

Minimality

If is another topology with:

then:

so:

Thus is smallest.

Uniqueness

If two smallest topologies existed, each would be contained in the other.

Hence they are equal.

Therefore the smallest topology exists and is unique.

This topology is called the topology generated by

Question 2

Consider with the standard topology.

(a) Prove

is open in .

In the standard topology on , open sets are unions of open intervals.

Since:

is an open interval, it is open.

(b) Is

open in ?

First compute the intersection.

If:

then:

Taking gives:

So:

But is not open in .

For any :

contains points other than .

Therefore:

is not open.

Hence:

is not open.

Key Concept

Topologies are closed under:

  • Arbitrary unions
  • Finite intersections

But not necessarily infinite intersections.

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