Path Connectedness

Section Path Connectedness

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As with every new property we define, it is natural to ask if path connectedness is a topological property.

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Activity 19.2.

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In this activity we prove Theorem 19.3.

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Theorem 19.3.

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Let

X

and

Y

be topological spaces and let

f : X \to Y

be a continuous function. If

A

is a path connected subspace of

X,

then

f (A)

is a path connected subspace of

Y .

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Assume that

X

and

Y

are topological spaces,

f : X \to Y

is a continuous function, and

A \subseteq X

is path connected. To prove that

f (A)

is path connected, we choose two elements

u

and

v

f (A) .

It follows that there exist elements

a

and

b

A

such that

f (a) = u

and

f (b) = v .

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(a)

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Explain why there is a continuous function

p : [0, 1] \to A

such that

p (0) = a

and

p (1) = b .

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(b)

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Determine how

p

and

f

can be used to define a path

q : [0, 1] \to f (A)

from

u

v .

Be sure to explain why

q

is a path. Conclude that

f (A)

is path connected.

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A consequence of Theorem 19.3 is the following.

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Corollary 19.4.

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Path connectedness is a topological property.

Topology: An Inquiry-Based Approach

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Section Path Connectedness

Activity 19.2.

Theorem 19.3.

(a)

(b)

Corollary 19.4.