Topology of decomposition of a space

Question

In pg 121 of this notes, the author outlines a construction of gluing bundles. The scenario begins with

Let $X= X_0 \cup X_1$ be union of two comapct spaces. $A = X_0 \cap X_1$ so that $X = X_0 \bigsqcup X_1 /A$.

Why does $X$ have the same topology as $X_0 \sqcup X_1 /A$?

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An open set in $X$ does map to an open set in RHS, but is the converse true?

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Answer 1

There is a surjection $X_0 \sqcup X_1 \to X$, which identify points in $A$. It induces a continuous bijection $(X_0 \sqcup X_1)/A \to X$ hence an homeomorphism since both spaces are compact.