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Math and science::Topology

Connectedness. 4 lemmas

1. The edge of a connected space

Let X be a topological space. Let A and B be subspaces of X with [some requirement].

If A is connected, then so is B.

2. [something of a something on] a connected space is connected

Let f:XY be a continuous map of topological spaces. If [something] then [something].

In particular, any quotient of a connected space is connected.

3. The [something] of two connected spaces is connected.

4. A space that has [a particular way of being composed] is connected.

Let X be a nonempty topological space and (Ai)iI a family of subspaces covering X. Suppose that Ai is [something] for each iI and that AiAj for each i,jI, then X is connected.

This lemma says that gluing together overlapping connected spaces produces connected spaces.