Let $X$ be a complex algebraic variety, possibly singular and/or non-compact. It is well known that if $X$ is smooth then its Euler characteristic is equal to its Euler characteristic with compact support: $\chi(X)=\chi_c(X)$.

Questions. (1) Is the same equality true if $X$ is singular? (I think it is still true for singular curves.)

(2) The proof I know in the smooth case uses Hironaka's theorem on resolution of singularities. Is there a more elementary proof?

Let $X$ be a complex algebraic variety, possibly singular and/or non-compact. It is well known that if $X$ is smooth then its Euler characteristic is equal to its Euler characteristic with compact support: $\chi(X)=\chi_c(X)$.

Questions. (1) Is the same equality true if $X$ is singular? (I think it is still true for singular curves.)

(2) Is there an elementary proof of this fact at least in the smooth case?

UPDATE: As mentioned by Reladenine Vakalwe below the proof in the smooth case follows immediately from the Poincare duality.