Path Connectedness and Connectedness

Section Path Connectedness and Connectedness

Path connectedness and connectedness are different concepts, but they are related. In this section we will show that any path connected space must also be connected. We will see later that the converse is not true except in finite topological spaces.

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Theorem 19.10.

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If a topological space

X

is path connected, then

X

is connected.

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Proof.

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Suppose that

X

is path connected. Let

a \in X

and for any

x \in X

let

p_{x}

be a path from

a

x .

Let

C_{x} = p_{x} ([0, 1]) .

Now

C_{x}

is the continuous image of the connected set

[0, 1]

R,

C_{x}

is connected. Also,

p_{x} (0) = a \in C_{x}

and

p_{x} (1) = x \in C_{x} .

Thus, every set

C_{x}

contains

a

and so

⋂_{x \in X} C_{x}

is not empty. Therefore,

C = ⋃_{x \in X} C_{x}

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is a connected subset of

X .

But every

x \in X

is in a

C_{x},

C = X .

We conclude that

X

is connected.

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In the following sections we explore the reverse implication in Theorem 19.10 — that is, does connectedness imply path connectedness.

Topology: An Inquiry-Based Approach

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Section Path Connectedness and Connectedness

Theorem 19.10.

Proof.