Worksheet 6

Question 1 — Unit Circle and Homeomorphisms

Let

denote the unit circle.

Determine whether is homeomorphic to:

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vs

is compact because it is a closed and bounded subset of .

is compact.

However:

Removing one point from disconnects it:

is disconnected.

But removing one point from leaves a connected set.

Since connectedness properties are preserved by homeomorphisms:

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vs

is compact but is not compact.

Compactness is preserved under homeomorphisms.

Therefore:

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Conclusion

is homeomorphic to neither nor .

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Question 2 — Images Under Homeomorphisms

Let

be a homeomorphism.

Let:

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(a) Closed Sets

Suppose is closed in .

Is:

closed in ?

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Proof

Since is a homeomorphism, is continuous.

We have:

Since is closed and is continuous:

is closed.

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Conclusion

Yes,

is closed.

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(b) Connected Sets

Suppose is connected.

Is:

connected?

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Proof

Continuous images of connected sets are connected.

Since is continuous:

is connected.

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Conclusion

Yes,

is connected.

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(c) Path Connected Sets

Suppose is path connected.

Is:

path connected?

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Proof

Let:

Then:

for some .

Since is path connected, there exists a continuous path:

with:

Define:

Then:

Thus is path connected.

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Conclusion

Yes,

is path connected.

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(d) Totally Disconnected Sets

Suppose is totally disconnected.

Is:

totally disconnected?

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Proof

Let be connected.

Then:

is connected in since is continuous.

Since is totally disconnected:

contains at most one point.

Thus:

contains at most one point.

Hence is totally disconnected.

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Conclusion

Yes,

is totally disconnected.

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(e) Compact Sets

Suppose is compact.

Is:

compact?

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Proof

Continuous images of compact sets are compact.

Since is continuous:

is compact.

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Conclusion

Yes,

is compact.