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I am a beginner in Topology and trying to learn by examples. I am finding problems in getting examples on intersection of open sets with other open sets or closed sets. Can someone please provide me with some good examples or links and books specially dealing with set theory dedicated to more of open sets. I am looking forward to concrete examples rather than symbolic manipulation.

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I wasn't going to upvote but since I see no reason for a downvote I am going to now. – Rudy the Reindeer Sep 26 '12 at 12:12
i feel glad that irrespective of votes I can ask my question !! – Rorschach Sep 26 '12 at 13:32
up vote 1 down vote accepted

Here are 2 examples of intersections of two open sets:

Consider $\mathbb R$ with the standard topology. Then $(0,1)$ and $(1,2)$ are open sets. $(0,1) \cap (1,2) = \varnothing$ is open, as expected.

Consider $\mathbb R$ with the cofinite topology. Then $\mathbb R \setminus \{1\}$ and $\mathbb R \setminus \{2,3,4\}$ are open and $\mathbb R \setminus \{1\}\cap \mathbb R \setminus \{2,3,4\} = \mathbb R \setminus \{1, 2,3,4\}$ is open again, as expected.

I learned topology from Mendelson which I read cover to cover. It's short (200 pages) and easy to read and comes with exercises at the end of each chapter. It's an excellent book.

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and in case of $\left(1,3 \right)\bigcap \left(2,3 \right]$ ? – Rorschach Sep 26 '12 at 12:04
@rafiki In this case you have $(1,3) \cap (2,3] = (2,3)$ which is also open in the standard topology. – Rudy the Reindeer Sep 26 '12 at 12:06

You can try this books:

1- Munkres. J. Topology

2- Dugundji. J. Topology

They are really good book, and you gonna finda a lot of examples.

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