Our chimney stack was solidly built from the ground up. It was a "channel" with two functions depending on the height of the channel in the chimney. From the basement up to the attic, the main pipe and the overflow pipe to the expansion tank ran through it. From the attic and up through the roof (in the same position), the intake went up, and they were separated by a stone thickness.
We tore down our brick chimney in 2009. Above the roof and in the attic, we dismantled it layer by layer. At ground level, we took down a section from the attic partway down, breaking many stones in the process. The reason we did this was to minimize dust inside. The chimney stack was plastered and had wallpaper on it at ground level, so we could then enclose half of the chimney stack and ourselves in a "tent" and dismantle the entire height at ground level and a little way down into the basement. In the basement, we intended to keep most of the outer layer.