What is the path space of a topological space X according to the video?

Conditions

• The ambient context is algebraic topology.
• X is a topological space.

Reasoning, step by step

1. Identify the set of maps from the interval to X.
2. Recognize that this set carries a topology.
3. Conclude that the path space is a topological space.

Example

The speaker says: 'As a set, it is the set of maps from the interval to X, but it's a very useful fact from topology that this is itself a topological space.'

Common misconceptions

• Thinking the path space is only a set of functions without topological structure.
• Assuming the topology on the path space is the discrete topology.

Watch the explanation

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• Why does the speaker emphasize that the path space is a topological space rather than only a set?
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