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- Homework Statement
- prove that a union of compact spaces is compact

- Relevant Equations
- o.o

Prove that if ##X## is a topological space, and ##S_i \subset X## is a finite collection of compact subspaces, then their union ##S_1 \cup \cdots \cup S_n## is also compact.

##S_i \subset X## is compact ##\therefore \forall S_i, \exists## a finite open cover ##\mathcal J_i=\{U_j\}_{j\in J\subset K} \subset \{U_k\}_{k\in K}=\mathcal K_i##, where ##\mathcal K_i## is an open cover of ##S_i##. I define a finite open cover ##\bigcup_i \mathcal J_i## of ##\bigcup_{i=1}^{n}S_i##. The fact that a finite union of finite sets is finite implies that ##\bigcup_i J_i## is a finite open cover of ##\bigcup_{i=1}^{n}S_i## ##\therefore \bigcup_{i=1}^{n}S_i## is compact by the definition of a compact topological space.

##S_i \subset X## is compact ##\therefore \forall S_i, \exists## a finite open cover ##\mathcal J_i=\{U_j\}_{j\in J\subset K} \subset \{U_k\}_{k\in K}=\mathcal K_i##, where ##\mathcal K_i## is an open cover of ##S_i##. I define a finite open cover ##\bigcup_i \mathcal J_i## of ##\bigcup_{i=1}^{n}S_i##. The fact that a finite union of finite sets is finite implies that ##\bigcup_i J_i## is a finite open cover of ##\bigcup_{i=1}^{n}S_i## ##\therefore \bigcup_{i=1}^{n}S_i## is compact by the definition of a compact topological space.