A topological property is a property that can be described by open sets. For example, continuity is a topological property because we can say that a function is continuous if every preimage of an open set is an open set. Topological properties are unchanged by a homeomorphism which is a continuous bijective function where the inverse is also continuous.
A metric space is a set $M$ equipped with a metric $d$ that satisfies positive definiteness, symmetry, and the triangle inequality. A metric subspace $N$ of $M$ is a metric space that inherits the metric $d$ from $M$.
The set $S = \{x \in \mathbb{Q}: -\pi < x < \pi \}$ is neither open nor closed in $\mathbb{R}$. To see why, recall the definition of open and closed sets. In an open set $S$, if $p \in S$ and $d(p,q) < \epsilon$ for some $\epsilon > 0,$ then $q \in S$. Since $S$ only contains elements in the rational numbers, there is an irrational number in every $\epsilon$-neighborhood that is not in $S$, so $S$ is not open in $\mathbb{R}$. For closed sets, every limit of $S$ should be included in $S$. This is not the case because $\pi$ is a limit point in $\mathbb{R}$, yet $\pi \notin S$. Thus, S is neither closed nor open in $\mathbb{R}$.
The same set $S$ is clopen in $\mathbb{Q}$. We take a brief detour to the inheritance principle which states that every subspace N of M inherits its topology from M in the sense that each open set $V \subset N$ is actually the intersection $V=N \cap U$ for some open set $U$ in $M$. It can also be shown that every metric subspace N of M inherits its closed sets from M by taking complements because the complement of a open set is a closed set. $\mathbb{Q}$ is a metric subspace of $\mathbb{R}$. Consider open set $O = (-\pi, \pi)$ and closed set $C = [-\pi, \pi]$ in $\mathbb{R}$. Now, $S = O \cap \mathbb{Q} = C \cap \mathbb{Q}$, so by the inheritance principle, $S$ is clopen.
Analysis is not very kind to me–there are so many c-words and I get them confused so easily! (Fun/sad fact: for a while in my analysis class, I didn’t realize continuity and convergence were different concepts) Just for the record, closed, convergent, Cauchy, compactness, connectedness, clustering, condensing, covering, cantor sets, completion, and so many proofs in analysis use contradiction.
Every convergent sequence is Cauchy, but not every Cauchy sequence is convergent. A metric space $M$ where any Cauchy sequence in M converges to an element in M is complete. $\mathbb{R}$ is complete while $\mathbb{Q}$ is not complete.
I remember learning about subsequences and not really understanding why they were important. Compactness is the why. A set $S$ is compact if for every sequence $(a_n)$, there exists a subsequence $(a_{n_k})$ that converges to an element in $S$. Why is compactness something of interest? How does it reduce infinite to finite?
To summarize, continuity, compactness, and connectedness are topological properties while completeness is not. If we are able to show that one metric space show that two metric spaces are not homeomorphic,