How to show induced topological space

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SUMMARY

The discussion focuses on the concept of induced topological spaces, specifically how to define a topology on a subset X of a topological space Y. The method involves defining open sets for X as the intersection of open sets U in Y with X, denoted as U ∩ X. The user initially struggled to prove that X is included in the collection of open sets defined in this manner but realized that by letting U equal Y, the proof becomes straightforward. The discussion highlights the importance of understanding the intersection properties of open sets in topology.

PREREQUISITES
  • Understanding of basic topology concepts, including open sets and topological spaces.
  • Familiarity with set operations, particularly intersection and union.
  • Knowledge of the notation used in topology, such as U ∩ X.
  • Basic experience with mathematical proofs and logical reasoning.
NEXT STEPS
  • Study the properties of induced topologies in more depth.
  • Learn about the concept of basis for a topology and how it relates to open sets.
  • Explore examples of topological spaces and their induced subspaces.
  • Investigate the role of closure and interior in the context of topological spaces.
USEFUL FOR

Students and educators in mathematics, particularly those focusing on topology, as well as researchers seeking to deepen their understanding of induced topological spaces and their properties.

ismaili
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I am beginning to read about the topology,
I met a problem puzzled me for a while.

If Y is a topological space, and X\subset Y, we can make the set X to be a topological space by defining the open set for it as U\cap X, where U is an open set of Y.

I would like to show that this indeed defines a topological space. But I failed to prove that there is the open set X among those open sets defined above, i.e. U\cap X. Anybody helps me?

Otherwise, we can easily see that
(U_1\cap X) \cap (U_2 \cap X) = (U_1\cap U_2)\cap X
and
(U_1\cap X) \cup (U_2\cap X) = (U_1\cup U_2) \cap X
and
\phi = \phi \cap X
And I lack the final piece that the X is contained in the collection of open sets of X defined above.

Thanks!
 
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Let U be Y.
 
George Jones said:
Let U be Y.

aha! How stupid I was!
Just let U be Y.
Thank you a lot. :shy:
 
Also, you might want to use [ itex] instead of [ tex ] if you don't want to start a new line every time...
 

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