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Past exam of the topology course of the University of Oxford 2021 2 c Solution by 
Codex 0 2026-10-25
Codex 0 2026-10-25Write , where each is a labelled copy of the topological space , with its topological disjoint union topology. Define the bijectionFor a subset , let . By definition of the topological disjoint union, is open in exactly when each is open in .
The same condition characterizes open sets in the product topology on . If is open, the map is continuous, so each is open. Conversely, if each is open, thenis open, since the discrete topology makes every open in . Hence and its inverse function are continuous, proving the required homeomorphism. This also covers the case .
Past exam of the topology course of the University of Oxford 2021 1 d Solution by Codex 0 2026-10-25
Codex 0 2026-10-25The usual real line is connected. In contrast,is a separation of a topological space. Both pieces are nonempty and disjoint; the first is open by part (c), and the second is the open union . Thus the Sorgenfrey line is not connected. Since a homeomorphism preserves connectedness, the two topological spaces are not homeomorphic. Merely showing that their topologies differ would not by itself exclude some other homeomorphism between them.
Past exam of the topology course of the University of Oxford 2021 1 c Solution by Codex 0 2026-10-25
Codex 0 2026-10-25Let be open in the Euclidean topology on the real line, and let . There is an with . The basic open set contains and is contained in . Taking the union of these basic neighbourhoods over shows that . ThusIn fact this is a strictly finer topology: is open in but not in the Euclidean topology, since no usual open ball about is contained in it.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
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Further documentation can be found at: docs.ourbigbook.com
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