StudyLabs

Worksheet 3

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Question 1 — Composition of Continuous Maps

Let , and be topological spaces.

Suppose

and

are continuous.

Prove that

is continuous.

Proof

Let be open.

Since is continuous:

is open in .

Since is continuous:

is open in .

But:

Therefore:

is continuous.

Question 2 — Inclusion Map

Let be a topological space and let .

Let be the subspace topology on .

Consider the inclusion map:

given by

Verify that is continuous.

Proof

Let be open.

Then:

Since is open in the subspace topology:

Therefore is continuous.

Question 3 — Discrete and Trivial Topologies

Let:

  • be the trivial topology on :
  • be the discrete topology on :

(a) Prove

is continuous.

Proof

Open sets in are:

Their preimages under id are:

Both are open in .

Therefore id is continuous.

(b) Suppose

is continuous. What can you say about ?

Proof

Continuity requires:

Every open set in must have open preimage in .

But:

for all .

So every subset must be open in .

But:

So the only subsets must be:

Thus:

Therefore:

has at most one element.

Question 4 — Translation on

Is translation on continuous?

Answer

Translation is the map:

where .

Each coordinate is:

Addition of real numbers is continuous.

Therefore:

is continuous.

Question 5 — Sum and Product Maps

Let

be defined by:

Let

be defined by:

Prove and are continuous.

Proof for S

Projection maps:

are continuous.

Addition:

is continuous.

Finite sums of continuous functions are continuous.

Hence:

is continuous.

Proof for P

Multiplication:

is continuous.

Finite products of continuous functions are continuous.

Thus:

is continuous.

Question 5(b) — Diagonal Map

Let

be:

Let:

be as above.

Is:

continuous?

Proof

Both maps are continuous:

is continuous because each coordinate map is .

is continuous from part (5).

Therefore:

is continuous.

Compute the Map

Polynomial Map

Consider:

given by:

Since:

and both maps are continuous:

is continuous.

Conclusion

All polynomial maps:

are continuous.

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