- The sequence criterion for closedness.
Let be a subset of . Then is closed if and only if the following condition holds:
for every sequence in which converges in , we have .
- The Bolzano–Weierstrass theorem
Every bounded sequence in has at least one convergent subsequence.
- Sequential compactness
Definition. Let be a subset of . Then is sequentially compact if every sequence in has at least one subsequence which converges and whose limit is in .
- The Heine–Borel Theorem (sequential compactness version) characterizing the sequentially compact subsets of as those which are both closed and bounded.
We always look at plenty of examples as we go (with more on the question sheets). However, when I have asked students to state these standard definitions or results in the past, I have found the following.
- Students have had no problem at all stating the Heine–Borel Theorem correctly.
- Many students have stated the sequence criterion for closedness incorrectly. The most common mistake is to claim that “a set E is closed if and only if every sequence in E converges to a point in E”.
- Many students have misquoted Bolzano–Weierstrass, claiming that “every bounded sequence converges”.
- Students are generally confused about why the ”if” in definitions behaves like an”if and only if”. (This is discussed in my G12MAN FAQ document)
- Many students have stated the definition of sequential compactness incorrectly. Here the variety of mistakes is very large! Common errors in defining the sequential compactness of a set E are to say one of the following:
- “every sequence in E converges to a point of E” .
(This is as above.)
- “every sequence in E which converges in must converge to a point of E”.
(This is the sequence criterion for closedness.)
- “every sequence in E has a subsequence which converges”
(Assuming that this means that the subsequence converges in , this is equivalent to the boundedness of E. Note that we do not treat subsets as spaces in their own right in this module.)
- “every convergent sequence in E has a subsequence which converges to a point of E”. (Again assuming this means convergent in , this is equivalent to the sequence criterion for closedness.)
I spent some extra time on this this year.
- I showed the students that the first of these is impossible if the set has at least two points, and so contrasted the correct and incorrect versions of the sequence criterion for closedness.
- I observed that statement 3 is equivalent to the boundedness of the set E, but is not enough to give sequential compactness. Here I illustrated the difference using the example of the open interval and the sequence .
Using this, I talked my way up through the different pieces of the definition of sequential compactness, explaining why each part of the definition adds more information.
Whether this has helped more of the students to understand this concept, I do not yet know!