Worksheet 4

Question 1 — Projection Maps in Product Topology

Let be a family of topological spaces.

Define

and

Let the projection maps be:

and

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Are the projection maps continuous?

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Finite Product Case

Let be open.

Then:

This is a basic open set in the product topology.

Therefore:

is continuous.

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Arbitrary Product Case

Let be open.

Then:

where:

This is open in the product topology.

Therefore:

is continuous.

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Question 2 — Homeomorphism Between Intervals

Find a homeomorphism between:

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Map

Define:

by

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Bijectivity

Inverse:

Thus is bijective.

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Continuity

is linear, hence continuous.

is also linear, hence continuous.

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Conclusion

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Question 3 — Is Homeomorphic to ?

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Map

Define:

by

(or equivalently)

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Properties

• Continuous
• Bijective
• Continuous inverse

Therefore:

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Question 4 — Matrix Spaces

Let

be the set of real matrices.

Identify:

Give the topology induced from .

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(a) Matrix Multiplication

Define:

by:

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Continuity

Each entry of is:

This is a polynomial in the matrix entries.

Since polynomials are continuous:

is continuous.

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(b) Determinant Map

Define:

by:

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Continuity

The determinant is:

This is a polynomial in the entries.

Therefore:

is continuous.

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Question 5 — General Linear Group

Define:

Prove:

is open in .

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Proof

We have:

Since:

is continuous and

is open in ,

its preimage is open.

Therefore:

is open.