(a) To check: If the given set forms a basis for a topology.

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Answer 1

Question

Chapter 4.36, Problem 5E

To determine

(a)

To check:

If the given set forms a basis for a topology.

To determine

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

To determine

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

To determine

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

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Answer 2

Question

Chapter 4.36, Problem 5E

To determine

(a)

To check:

If the given set forms a basis for a topology.

To determine

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

To determine

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

To determine

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Expert Solution & Answer

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Answer 3

Question

Chapter 4.36, Problem 5E

To determine

(a)

To check:

If the given set forms a basis for a topology.

To determine

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

To determine

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

To determine

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 4.36, Problem 5E

To determine

(a)

To check:

If the given set forms a basis for a topology.

To determine

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

To determine

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

To determine

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 5

Chapter 4.36, Problem 5E

To determine

(a)

To check:

If the given set forms a basis for a topology.

To determine

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

To determine

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

To determine

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Answer 6

Chapter 4.36, Problem 5E

To determine

(a)

To check:

If the given set forms a basis for a topology.

Answer 7

Chapter 4.36, Problem 5E

To determine

(a)

To check:

If the given set forms a basis for a topology.

Answer 8

To determine

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

Answer 9

To determine

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

Answer 10

To determine

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

Answer 11

To determine

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

Answer 12

To determine

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Answer 13

To determine

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Answer 14

Expert Solution & Answer

Answer 15

Expert Solution & Answer

Answer 16

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Answer 17

Ch. 4.30 - Show that l and I02 are not metrizable.Ch. 4.30 - Which of our four countability axioms does S...Ch. 4.30 - Which of our four countability axioms does in the...Ch. 4.30 - Let A be a closed subspace of X. Show that if X is...Ch. 4.30 - Prob. 10E Ch. 4.30 - Let f:XY be continuous. Show that if X is...Ch. 4.30 - Let f:XY be continuous open map. Show that if X...Ch. 4.30 - Show that if X has a countable dense subset, every...Ch. 4.30 - Show that if X is Lindelof and Y is compact, then...Ch. 4.30 - Give I the uniform metric, where I=[0,1]. Let...

(a) To check: If the given set forms a basis for a topology.

Solution Summary: The author explains how X forms a basis for the topology as shown below. The set is given by R-left0rightcup

Concept explainers

(a)

To check:

If the given set forms a basis for a topology.

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Want to see the full answer?

Chapter 4 Solutions

(a) To check: If the given set forms a basis for a topology.

Solution Summary: The author explains how X forms a basis for the topology as shown below. The set is given by R-left0rightcup

Concept explainers

(a) To check: If the given set forms a basis for a topology.

(b) To show: Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

(c) To show: The space X satisfies the T1 axiom, but is not Hausdorff.

(d) To show: The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.

Want to see the full answer?

Chapter 4 Solutions

(a)

To check:

If the given set forms a basis for a topology.

(b)

To show:

Each of the spaces X−{p} and X−{q} is homeomorphic to ℝ.

(c)

To show:

The space X satisfies the T1 axiom, but is not Hausdorff.

(d)

To show:

The space X satisfies all the conditions for a 1-manifold except for the Hausdorff condition.