T2-ness property is inherited in subspaces and in the topological product. What about T1 and T0, are they inheritable? Can someone give simple examples when this happens and when this does not happen, ie of Ti being inheritable , not inheritable in both the product and the subspace topology for i=0,1?
2026-03-26 02:56:12.1774493772
Are T1 and T0 inheritable in the subspace and product topologies?
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Hint: For both properties, you can show that they are inherited by subspaces and products immediately using the definition (for product, use the fact that if two points are distinct, then they differ on a coordinate).