Topology basics would tell me that fundamentally at least as far as a simple overview goes... a “black hole” doesn’t imply a topological change however a “wormhole” does imply a topological change. One “punches through” the otherwise “flat” space time, creating another “opening” in the surface and increasing the complexity of that space time surface.
To use the proper topology related words... The presence of a wormhole would make a region of space time that contained both ends of that wormhole, non-homeomorphic with that same region of space time without the wormhole. A black hole would not affect how homeomorphic the two regions of space time are, therefore it’s not really a change in the “topology” of the space even if the black hole would definitely change the geometry of that space quite drastically.
To use the proper topology related words... The presence of a wormhole would make a region of space time that contained both ends of that wormhole, non-homeomorphic with that same region of space time without the wormhole. A black hole would not affect how homeomorphic the two regions of space time are, therefore it’s not really a change in the “topology” of the space even if the black hole would definitely change the geometry of that space quite drastically.