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content/Definitions/Topological Spaces/Terminologies/Compact.md Normal file
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# Definition
$X$ is **compact** if every [[Open Cover|open cover]] has a finite [[Subcover|subcover]].

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content/Definitions/Topological Spaces/Terminologies/Connected Component.md Normal file
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# Definition
A **connected component** of a [[Topological Space|topological space]] is the union of all [[Connected|connected]] subsets that contain a given point. It itself is [[Connected|connected]].

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# Definition
A [[Topological Space|topological space]] is **connected** if it is not a union of two non-empty [[Open Sets|open sets]].
i.e. if you draw the two non-empty [[Open Sets|open sets]] on the graph, if you have to lift your pen, it will not be connected.
# Examples
Not connected:
$$\langle 0, 1 ] \cup [ 2, 5 \rangle$$
Connected:
$$\langle 0, 1 ] \cup [ 0.5, 5 \rangle = \langle 0, 5 \rangle$$