Basic Topology
Math
Anki notes for Topology.
Follows the notes by Tom Leinster: https://www.maths.ed.ac.uk/~tl/topology/topology_notes.pdf
A work in progress; some cards are marked as "TODO".
Shared with permission from Tom Leinster.
Sample Data
| Title | Connectedness, compactness and some fundamental theorems of calculus |
| Text | The following three theorems in calculus, theorems about functions from and to the reals, have generalizations in topology. Intermediate value theorem If \( f : [a, b] \to \mathbb{R} \) is continuous, and if \( r \) is a real number between \( f(a) \) and \( f(b) \), then {{c1::there exists an element \( c \in [a, b] \) such that \( f(c) = r \)}}. Maximum value theorem If \( f : [a, b] \to \mathbb{R} \) is continuous, then {{c1::there exists an element \( c \in [a, b] \) such that \( f(x) \le f(c) \) for every \( x \in [a, b] \)}}. Uniform continuity theorem If \( f : [a, b] \to \mathbb{R} \) is continuous, then for every \( \varepsilon > 0 \) {{c1::there exists a \( \delta > 0 \) such that \( |f(x_1) - f(x_2)| < \varepsilon \) for every \( x_1, x_2 \in [a, b] \) for which \( |x_1 - x_2| < \delta \)}}. Applications in Calculus The intermediate value theorem is used for constructing inverse functions, such as \( \sqrt[3]{x} \) and \( \arcsin(x) \). The maximum value theorem is used to prove the mean value theorem for derivatives, which in turn is used to prove the two fundamental theorems of calculus. The uniform continuity theorem is used for proving that every continuous function is integrable. What is the concept in question: functions vs sets? The three theorems can be considered to be describing facts about continuous functions; but shifting one's focus, one can view them as describing the nature of {{c1::the closed interval \( [a, b] \subset \mathbb{R} \)}}. As topological properties The topological property of the space \( [a, b] \) on which the intermediate value theorem depends is the topological property called {{c1::connectedness}}. The property which the maximum value theorem and the uniform continuity theorem depend on is called {{c1::compactness}}. Both of these properties are fundamental to areas beyond calculus; they are fundamental to almost any area which can be represented in topology. |
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| Example | |
| Source | Munkresp145 |
| Order | 4.1.1 |
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| Title | Compactness |
| Text | We use the idea of a cover and subcover to formulate a definition of compactness. Cover Let \( X \) be a topological space. A cover of \( X \) is a family \( (U_i)_{i \in I} \) of subsets of \( X \) such that {{c1::\( \bigcup_{i \in I} U_i = X \)}}. It is finite iff the indexing set \( I \) is finite, and open iff \( U_i \) is open for each \( i \in I \). Given a cover \( (U_i)_{i \in I} \) and \( J \subseteq I \), we say that {{c1::\( (U_j)_{j \in J} \)}} is a subcover of \( (U_i)_{i \in I} \) if it is itself a cover of \( X \). Compact A topological space \( X \) is compact iff {{c1::every open cover of \( X \) has a finite subcover}}. |
| BackOnlyText | Equivalent definitions There are multiple ways of formulating compactness, with some being more intuitive than others. The first alternative is worded specifically for the reals (is there a generalized version?) Compact (sequences in \( \mathbb{R} \)) A set \( K \subset \mathbb{R} \) is compact iff every sequence in \( K \) has a subsequences that converges to a limit that is also in \( K \). The most basic example of such a closed set in \( \mathbb{R} \) is a closed interval. Compact (closed, bounded sets) A set \( K \subset \mathbb{R} \) is compact iff it is closed and bounded. Heine-Borel theorem The Heine-Borel theorem is precicely the statement that ties these three notions of compactness together. Heine-Borel theorem Let \( K \) be a subset of \( \mathbb{R} \). The following three statements are equivalent: \( K \) is compact. \( K \) is closed and bounded. Every open cover of \( K \) has a finite subcover. The essence of compactness Abbott on p97 mentions this about compact sets and closed intervals: There may be a temptation to consider closed intervals as being a kind of standard archetype for compact sets, but this is misleading. The structure of compact sets can be much more intricate and interesting. For instance, one implication of Theorem 3.3.4 is that the Cantor set is compact. It is more useful to think of compact sets as generalizations of closed intervals. When ever a fact involving closed intervals is true, it is often the case that the same result holds when we replace “closed interval” with “compact set.” As an example, let’s experiment with the Nested Interval Property proved in the first chapter... He then goes on to prove a version of the nested closed interval property, but with compact sets replacing closed intervals: If \( K_1 \supseteq K_2 \supseteq K_3 \supseteq K_4 \supseteq ... \) is a nested sequence of nonempty compact sets, then the intersection \( \cap_{n=1}^{\infty} K_n \) is not empty. |
| Example | |
| Source | Abbott p97 |
| Order | 2.2.2 |
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| Title | Hausdorff spaces |
| Text | Most interesting topological spaces are Hausdorff. Hausdorffness is identified as being the second separation condition. The first condition, T1 is a sub-requirement of T2 (Housdorff), so it is useful to keep it in mind when thinking about the Hausdorff condition. T1 A topological space \( X \) is said to be \( T_1 \) iff {{c1::every one-element subset of \( X \) is closed}}. Now for Hausdorff. Hausdorff A topological space \( X \) is Hausdorff (or \( T_2 \)) iff {{c1::for every distinct \( x, y \in X \), there exists disjoint neighbourhoods of \( x \) and \( y \)}}. Lemma. Every Hausdorff space is \( T_1 \). |
| BackOnlyText | Slightly more precise wording of the Hausdorff condition:
A topological space \( X \) is Hausdorff (or \( T_2 \)) iff for every
\( x, y \in X \) such that \( x \ne y \), there exists disjoint open sets
\( U, W \) of \( X \) such that \( x \in U \) and \( y \in W \).
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| Example | Every metrizable space is Hausdorff. While most interesting spaces are Hausdorff, there are some non-Hausdorff spaces that are important. The Zariski topology is an example. |
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| Order | 1.1.11 |
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