Why always close Database connection?

If connecting to a database consumes a lot of resources, why should a database connection always be closed in your application if you have to open it again? Can I just make this connection available globally throughout my application so that other classes and methods can reuse it?

For example (in pseudo code):

public class PopulateGridViews()
{
    public SqlConnection conn = new SqlConnection(@"Database:DATABASE");
    conn.Open();

    void PopulateGrid1()
    {
        SqlCommand cmd = new SqlCommand("SELECT * FROM TABLE1");
        cmd.Connection = conn;
        cmd.ExecuteNonQuery();
        cmd.Dispose();
        // Populate Grid1
    }

    void PopulateGrid2()
    {
        SqlCommand cmd = new SqlCommand("SELECT * FROM TABLE2");
        cmd.Connection = conn;
        cmd.ExecuteNonQuery();
        cmd.Dispose();
        // Populate Grid2
    }
}