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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. A [something] of a connected space is connected

Let $$f : X \to Y$$ be a continuous map of topological spaces. If [something] then [something].

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

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

Let $$X$$ be a nonempty topological space and $$(A_i)_{i \in I}$$ a family of subspaces covering $$X$$. Suppose that $$A_i$$ is [something] for each $$i \in I$$ and that $$A_i \cap A_j \neq \emptyset$$ for each $$i, j \in I$$, then $$X$$ is connected.

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