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## Homework Statement

Suppose that [itex](X,\tau)[/itex] is the co-finite topological space on X.

I : Suppose A is a finite subset of X, show that [itex](A,\tau)[/itex] is discrete topological space on A.

II : Suppose A is an infinite subset of X, show that [itex](A,\tau)[/itex] inherits co-finite topology from [itex](X,\tau)[/itex].

## The Attempt at a Solution

I: Well, first I show that [itex]\forall a\in A: (A^c \cup \{a\})\in \tau[/itex]. I need to show that [itex] (A^c \cup \{a\})^c [/itex] is finite because I want it to be in [itex]\tau[/itex]. But:

[itex] (A^c \cup \{a\})^c = A \cap (X-\{a\}) = A - \{a\}[/itex], but since A is finite, if we remove one of its element, the remaining set is again finite. Therefore [itex]\forall a\in A: (A^c \cup \{a\})\in \tau[/itex]

Now since all the open sets in the subspace topology are the intersection of [itex]A[/itex] with something in [itex]\tau[/itex], we see that [itex]\{a\} = A \cap (A^c \cup \{a\})[/itex] is open in [itex](A,\tau)[/itex]. That means the topology induced on A will be the discrete topology.

II: What should I do for the second part? Any helps would be appreciated.