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Urysohn's lemma

Support

The support of complex function \(f\) on a topological space \(X\) is a closer of the set \({x: f(x) \neq 0}\)

\(C_c(X)\) denotes the collection of all continuous complex function whose support is compact.
\(C_c(X)\) is vector space because \(C_c(X)\) is closed under addition and multiplication.

\(K\prec f\) denotes the \(K\) is compact subset of X, that \(f \in C_c(X)\) that \(0 \leq f(X) \leq 1\) for all \(x \in X\) and that \(f(X) = 1\) for all \(x \in K\)

\(f \prec V\) denotes the \(V\) is open subset of X, that \(f \in C_c(X)\) that \(0 \leq f(X) \leq 1\) for all \(x \in X\) and that support of \(f\) lies in V

Urysohn’s lemma

Suppose \(X\) is a locally compact Hausdorff space, \(V\) is open in \(X\), \(K \subset X\) and \(K\) is compact then there is \(C_c(X)\) such that \(K \prec f \prec V\) or \(\chi_K \leq f \leq \chi_V\)

Put \(r_1 = 0, r_2 = 1\) and let \(r_3, r_4 ...\) be enumulation of rationals in (0,1).