It is a consequence of Sullivan's work

Sullivan, D., Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. 192, 165-168 (1971). ZBL0227.32005.

that every compact $k$-dimensional real-analytic subset $V\subset {\mathbb R}^n$ is a mod 2 pseudo-manifold: Every $k-1$-dimensional simplex in a triangulation of $V$ is contained in an even number of $k$-simplices. (This is immediate from the main result of Sullivan's paper about local structure of $V$ as a cone over a base of even Euler characteristic.) Now, take the sum of all $k$-simplices in the given triangulation of $V$. This will be a mod 2 cycle. Since there are no simplces of dimension $k+1$, this cycle is not a boundary. Hence, $H_k(V, {\mathbb Z}_2)\ne 0$. In particular, $V$ cannot be contractible. Note that this argument is pretty much the same as in the smooth case, one you have Sullivan's local result.

Sullivan's paper is freely available here.

It is a consequence of Sullivan's work

Sullivan, D., Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. 192, 165-168 (1971). ZBL0227.32005.

that every compact $k$-dimensional real-analytic subset $V\subset {\mathbb R}^n$ is a mod 2 pseudo-manifold: Every $k-1$-dimensional simplex in a triangulation of $V$ is contained in an even number of $k$-simplices. (This is immediate from the main result of Sullivan's paper about local structure of $V$ as a cone over a base of even Euler characteristic.) Now, take the sum of all $k$-simplices in the given triangulation of $V$. This will be a mod 2 cycle. Since there are no simplices of dimension $k+1$, this cycle is not a boundary. Hence, $H_k(V, {\mathbb Z}_2)\ne 0$. In particular, $V$ cannot be contractible. Note that this argument is pretty much the same as in the smooth case, once you have Sullivan's local result.

Sullivan's paper is freely available here.

It is a consequence of Sullivan's work

Sullivan, D., Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. 192, 165-168 (1971). ZBL0227.32005.

that every compact real-analytic subset $V\subset {\mathbb R}^n$ is a mod 2 pseudo-manifold, hence, has a ${\mathbb Z}_2$-fundamental class, hence, has nonzero $H_k(V, {\mathbb Z}_2)$, $k=\dim(V)$. In particular, $V$ cannot be contractible.

Sullivan's paper is freely available here.

It is a consequence of Sullivan's work

Sullivan, D., Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. 192, 165-168 (1971). ZBL0227.32005.

that every compact $k$-dimensional real-analytic subset $V\subset {\mathbb R}^n$ is a mod 2 pseudo-manifold: Every $k-1$-dimensional simplex in a triangulation of $V$ is contained in an even number of $k$-simplices. (This is immediate from the main result of Sullivan's paper about local structure of $V$ as a cone over a base of even Euler characteristic.) Now, take the sum of all $k$-simplices in the given triangulation of $V$. This will be a mod 2 cycle. Since there are no simplces of dimension $k+1$, this cycle is not a boundary. Hence, $H_k(V, {\mathbb Z}_2)\ne 0$. In particular, $V$ cannot be contractible. Note that this argument is pretty much the same as in the smooth case, one you have Sullivan's local result.

Sullivan's paper is freely available here.