How Do Subspace Topologies Compare When One is Finer?

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In summary, the subspace topology induced by a finer topology on a subset of X is always finer than the subspace topology induced by the original topology, but it may not be strictly finer in all cases. This can be shown by re-reading the definition of subspace topology and considering a single point as the subset Y.
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tomboi03
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If T and T' are topologies on X and T' is strictly finer than T, what can you say about the corresponding subspace topologies on the subset Y of X?

See the thing is... I know that they CAN be finer but that's might not be for every case. because Y is JUST a subset of X. This does not necessarily mean finer or coarser.

How do i put this in terms of symbols in my proof?
 
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I would reread the definition of subspace topology.

For your question, suppose A is open in the subspace topology induced by T. Then by definition, there exists U open in X such that A = U int Y. Since U is also open in the T' topology this shows that A is open in the topology induced by T'. This shows that the T' induced topology is finer than the T-induced topology.

But it needn't be strictly finer (Hint: let Y be a single point).
 

FAQ: How Do Subspace Topologies Compare When One is Finer?

What is the difference between a finer and coarser topology?

A finer topology refers to a topology that has more open sets than another topology, while a coarser topology has fewer open sets. In other words, a finer topology is more detailed and specific, while a coarser topology is more general and less specific.

How do finer and coarser topologies affect the structure of a space?

Finer and coarser topologies can change the way a space is structured by altering which points are considered to be in the same neighborhood. In a finer topology, two points may be considered closer together because there are more open sets that contain them, while in a coarser topology, those same points may not be considered close because there are fewer open sets that contain them.

Can a topology be both finer and coarser than another topology?

No, a topology cannot be both finer and coarser than another topology. A topology can only be either finer or coarser, and the comparison depends on the number of open sets in each topology.

What is the relationship between finer and coarser topologies?

A finer topology is said to be a refinement of a coarser topology, and a coarser topology is said to be a coarsening of a finer topology. This means that a finer topology contains all the same open sets as a coarser topology, in addition to some extra open sets.

How do finer and coarser topologies affect the continuity of a function?

A finer topology can make a function seem discontinuous because the function may not be continuous with respect to the additional open sets. On the other hand, a coarser topology can make a function appear continuous because the function may be continuous with respect to the fewer open sets. The choice of topology can greatly impact the continuity of a function.

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