Worksheet 5

Question 1 — Orthogonal Group is Closed

Define

Here:

• is the transpose of
• is the identity matrix

Prove:

is closed in .

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Proof

Define:

by

Matrix multiplication and transpose are continuous maps, so:

is continuous.

Then:

Since is closed in and is continuous, the preimage of a closed set is closed.

Therefore:

is closed.

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Question 2 — Special Orthogonal Group

Define:

Prove:

is closed in .

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Proof

We know:

is closed in .

Also:

Since:

• determinant is continuous
• is closed in

we get:

is closed.

Intersection of closed sets is closed, therefore:

is closed.

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Question 3 — Closure in and

Let with .

Define:

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(a) Closure in

The closure is:

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Justification

Every rational number between and is already in .

Also, rationals can approximate and arbitrarily closely.

Thus:

belong to the closure whenever they are rational.

Hence:

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(b) Closure in

Now consider as a subset of .

Then:

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Justification

Rational numbers are dense in .

Every real number in is a limit of rational numbers in .

Therefore:

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Question 4 — Distance to a Set

Let be a metric space.

Let:

be open and

be closed.

Define:

and

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(a) Is continuous?

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Proof

For any :

and

By triangle inequality:

Taking infimum:

Similarly:

Thus:

Therefore is Lipschitz and hence continuous.

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(b) Is continuous?

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Proof

Similarly:

Using the same argument:

Thus is Lipschitz and therefore continuous.

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Conclusion

Both functions:

are continuous on .