How to show induced topological space

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Discussion Overview

The discussion revolves around the concept of induced topological spaces, specifically how to show that a subset X of a topological space Y can be equipped with a topology defined by the intersection of open sets in Y with X. The focus is on proving that this definition satisfies the axioms of a topological space.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant describes the process of defining a topology on X as the intersection of open sets U in Y with X, seeking to prove that this forms a valid topological space.
  • The same participant notes that they can demonstrate closure under intersections and unions but struggles to show that X itself is included in the collection of open sets defined.
  • Another participant suggests simply letting U be Y, which appears to resolve the initial participant's confusion.
  • A later reply offers a suggestion regarding the use of different LaTeX formatting for mathematical expressions.

Areas of Agreement / Disagreement

Participants do not appear to disagree on the fundamental concept but rather engage in a clarification process, with one participant expressing realization and gratitude for the suggestion provided.

Contextual Notes

The discussion does not fully resolve the initial participant's concern about the inclusion of X in the collection of open sets, as it primarily shifts focus to a simpler case without addressing all axioms of a topological space.

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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