Connectedness and components
Why this is asked: Components are always closed but not always open; in ℚ every component is a single point. Use clopen sets or a continuous surjection onto {0,1} to disprove connectedness.
Connectedness, components and totally disconnected spaces
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The trap here
“Connected components are open” — false
with the usual topology
Components are singletons, which are not open. They are open exactly when the space is locally connected.
Next: Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set
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Open this in the full syllabus view · Unit 2