# 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).