How to show induced topological space

[ismaili]

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!

 

[George Jones]

Let U be Y.

 

[ismaili]

Let U be Y.

aha! How stupid I was!
Just let $$U$$ be $$Y$$.
Thank you a lot. :shy:

 

[micromass]

Also, you might want to use [ itex] instead of [ tex ] if you don't want to start a new line every time...

 

[blue_raver22]

Dear reader,

Thank you for reaching out with your question about induced topological spaces. I can understand your confusion, as topology can be a difficult subject to grasp at first. However, I am here to help guide you through the concept of induced topological spaces and how to show that it is indeed a topological space.

Firstly, let's define what an induced topological space is. An induced topological space is a topological space that is created from a subset of a larger topological space. In other words, if we have a topological space Y and a subset X of Y, we can define a new topological space by considering only the open sets of Y that intersect with X.

To show that this indeed defines a topological space, we need to prove three things: 1) the empty set and the entire X set are open sets, 2) the intersection of any two open sets is also an open set, and 3) the union of any collection of open sets is also an open set.

1) The empty set and the entire X set are open sets:
The empty set is defined as the set that contains no elements. In this case, the empty set is the intersection of the empty set and X, which is still the empty set. Therefore, the empty set is an open set in the induced topological space.

Similarly, the entire X set is the intersection of the entire Y set and X, which is still the entire X set. Therefore, the entire X set is also an open set in the induced topological space.

2) The intersection of any two open sets is also an open set:
Let U and V be two open sets in the induced topological space. This means that U = U_1 \cap X and V = U_2 \cap X, where U_1 and U_2 are open sets in the larger topological space Y.

Now, the intersection of U and V can be written as (U_1 \cap X) \cap (U_2 \cap X). Using the associative property of intersections, we can rewrite this as (U_1 \cap U_2) \cap X. Since U_1 and U_2 are open sets in Y, their intersection U_1 \cap U_2 is also an open set in Y. Therefore, (U_1 \cap U_2) \cap X is an open set in the induced