Basic Topology

Metric Spaces

Note

• Metric Space: A set \(X\) is a metric space if there exists a function \(d : X \times X \to \mathbb{R}_0^{+}\) such that for any \(p,q,r \in X\):

  1. \(d(p,q) = 0 \Longleftrightarrow p = q\)

  2. \(d(p,q) = d(q,p)\)

  3. \(d(p,q) \le d(p,r) + d(r,q)\)

  The function \(d\) is called a metric/distance-function.

Note

• Points: The elements of a metric space are called points.

Useful definitions for Euclidean Space

Note

• Segment and Interval in :math:`mathbb{R}`: For \(a,b \in \mathbb{R}\) with \(a<b\),

  \[(a,b) := \{x \mid a < x < b\}\]

  \[[a,b] := \{x \mid a \le x \le b\}.\]

Note

• k-cells in :math:`mathbb{R}^k`: For \(a_i,b_i \in \mathbb{R}\) with \(a_i < b_i\) for all \(i \in \{1,\dots,k\}\), a \(k\)-cell is

  \[\{x \mid a_i \le x_i \le b_i\}\]

  where \(x = (x_1,\cdots,x_k)^{T} \in \mathbb{R}^k\).

Note

• Open/Closed-Balls in :math:`mathbb{R}^k`: For \(x \in \mathbb{R}^k\), an open ball (or closed ball) centered at \(x\) with radius \(r>0\) is

  \[B_r(x) := \{y \in \mathbb{R}^k \mid |x-y| < r\}\]

  \[\bar{B}_r(x) := \{y \in \mathbb{R}^k \mid |x-y| \le r\}.\]

Note

• Convex Sets in :math:`mathbb{R}^k`: \(E \subseteq \mathbb{R}^k\) is convex iff \(\forall x,y \in E\) and \(\forall \lambda \in (0,1)\),

  \[\lambda x + (1-\lambda)y \in E.\]

Note

• Theorem: \(k\)-cells and open/closed balls are convex sets in \(\mathbb{R}^k\).

Useful definitions for General Metric Space

In all the following definitions, \(X\) is a metric space.

Note

• Neighbourhood: For \(p \in X\) and \(r>0\),

  \[N_r(p) := \{q \in X \mid d(p,q) < r\}.\]

Note

• Bounded Set: \(E \subseteq X\) is bounded iff \(\exists p \in X\) and \(\exists M>0\) such that \(E \subseteq N_M(p)\).

Note

• Limit Points: \(p \in X\) is a limit-point of \(E \subseteq X\) iff every neighbourhood of \(p\) contains at least one point from \(E\) other than \(p\) itself:

  \[\forall r>0,\ \exists q \ne p \in N_r(p) \text{ such that } q \in E.\]

Tip

Limit point of a set \(E\) (which may or may not be inside \(E\)) means we can get arbitrarily close to that point without stepping outside of \(E\). Limit points also form the border of \(E\).

Note

• Interior Points: \(p \in E\) is an interior point of \(E\) iff \(\exists r>0\) such that \(N_r(p) \subseteq E\).

Tip

Interior point of a set \(E\) (which has to be inside \(E\)) means we have a little breathing space to step outside of that point without crossing the border of \(E\).

Note

• Isolated Points: All points \(p \in E\) that are not limit points of \(E\) (\(p \in E \wedge p \notin E^{\uparrow}\)) are isolated points of \(E\).

Tip

Can’t get close to those points without crossing \(E\)’s border.

Note

• Closed Set: If \(E \subseteq X\) contains all of its own limit points (\(p \in E^{\uparrow} \implies p \in E\)) it is closed.

Tip

All of \(E\)’s border belongs to \(E\).

Note

• Perfect Set: If all points of a closed set \(E\) are limit points (\(p \in E \implies p \in E^{\uparrow}\)), then \(E\) is perfect.

Tip

Every point in \(E\) is reachable without stepping outside of \(E\).

Note

• Open Set: If all points of \(E\) are interior points, then \(E\) is open. Symbolically,

  \[p \in E \Longrightarrow p \in E^{0}.\]

Tip

Every point in \(E\) has a little breathing space around it within \(E\).

Note

• Closure: The set

  \[E^{\circ} := E \cup E^{\uparrow}\]

  is the closure of \(E\).

Note

• Dense Set: A set \(E\) is dense in \(X\) iff every point in \(X\) lies within the closure of \(E\) (\(p \in X \Longrightarrow p \in E^{\circ}\)).

Properties involving Metric Spaces

Note

• Theorem: Every neighbourhood is an open set.

• Theorem: If \(p\) is a limit point of \(E\), then every neighbourhood of \(p\) contains infinitely many points of \(E\).

• Corollary: A finite set has no limit points.

• Theorem: \(F\) is closed \(\Longleftrightarrow F^{c}\) is open.

• Theorem (open/closed sets):

  1. For a collection of open sets \(\{G_{\alpha}\}\), \(\bigcup_{\alpha} G_{\alpha}\) is open.

  2. For a collection of closed sets \(\{F_{\alpha}\}\), \(\bigcap_{\alpha} F_{\alpha}\) is closed.

  3. For a finite collection of open sets \(\{G_i\}_{i=1}^{n}\), \(\bigcap_{i=1}^{n} G_i\) is open.

  4. For a finite collection of closed sets \(\{F_i\}_{i=1}^{n}\), \(\bigcup_{i=1}^{n} F_i\) is closed.

• Theorem: For \(E \subseteq X\):

  1. \(E^{\circ}\) is closed.

  2. \(E = E^{\circ}\) iff \(E\) is closed.

  3. For every closed set \(F \subseteq X\), \(E \subseteq F \Longrightarrow E^{\circ} \subseteq F\).

• Theorem: Let \(E\) be a nonempty set of real numbers which is bounded above. Let \(y=\sup E\). Then \(y \in E^{\circ}\) (and \(y \in E\), if \(E\) is closed).

Note

• Relatively Open: Suppose \(E \subset Y \subset X\). \(E\) is open relative to \(Y\) if \(\forall p \in E\), \(\exists r>0\) such that \(\exists q \in Y\) with \(d(p,q) < r \Longrightarrow q \in E\).

Note

• Example: A set \(E\) can be relatively open to \(Y\) without being open in \(X\). For example, the segment \((a,b)\) is not open in \(\mathbb{R}^{2}\) (any neighbourhood defined on \(\mathbb{R}^{2}\) around a point in the segment ought to contain points from the orthogonal direction - which are not in the segment). However, it is relatively open to \(\mathbb{R}\).

Note

• Theorem: For \(E \subset Y \subset X\), \(E\) is open relative to \(Y \Longleftrightarrow E = Y \cap G\) for some open set \(G \subset X\).

Compact Sets

Useful Definitions

In all the following definitions, \(X\) is a metric space.

Note

• Open Cover: A collection of open sets \(\{G_{\alpha}\}\) is an open cover of \(E \subseteq X\) iff

  \[E \subset \bigcup_{\alpha \in A} G_{\alpha}\]

  for some index set \(A\).

Note

• Compactness: \(K \subseteq X\) is compact iff every open cover of \(K\) contains a finite sub-cover. Formally, for any open cover of \(K\), \(\{G_{\alpha}\}\), \(\exists \alpha_1,\cdots,\alpha_n\) for some \(n>0\) such that

  \[K \subset \bigcup_{i=1}^{n} G_{\alpha_i}.\]

Note

• Sequential Compactness: \(K \subset X\) is sequentially compact if every infinite subset \(E \subset K\) has a limit point in \(K\), or alternatively, every sequence in \(K\) has a subsequence which converges to some point in \(K\).

Properties Related to General Metric Space

Note

• Theorem: For \(K \subset Y \subset X\), \(K\) is compact relative to \(X \Longleftrightarrow K\) is compact relative to \(Y\).

Note

• Theorem: Every compact set is closed.

• Theorem: Every closed subset of a compact set is compact.

• Corollary: If \(F \subset X\) is closed and \(K \subset X\) is compact then \(F \cap K\) is compact.

• Theorem: For a collection of compact sets \(\{K_{\alpha}\}\), if \(\bigcap_{i=1}^{n} K_{\alpha_i} \ne \varnothing\) for every finite subcollection \(\{K_{\alpha_i}\}_{i=1}^{n}\), then

  \[\bigcap_{\alpha} K_{\alpha} \ne \varnothing.\]

• Corollary: For a sequence of non-empty compact sets \(\{K_n\}\), \(\forall n>0,\ K_n \supset K_{n+1} \Longrightarrow \bigcap_{i=1}^{\infty} K_i \ne \varnothing\).

• Theorem: For metric spaces, compactness \(\Longleftrightarrow\) sequential-compactness.

Properties Related to Euclidean Space

Note

• Theorem (\(\mathbb{R}\)): For a sequence of intervals \(\{I_n\}\) such that \(I_n \supset I_{n+1}\) for all \(n>0\), we have \(\bigcap_{i=1}^{\infty} I_i \ne \varnothing\).

Note

• Theorem (\(\mathbb{R}^k\)): For a sequence of \(k\)-cells \(\{I_n\}\) such that \(I_n \supset I_{n+1}\) for all \(n>0\), we have \(\bigcap_{i=1}^{\infty} I_i \ne \varnothing\).

Note

• Theorem (\(\mathbb{R}^k\)): Every \(k\)-cell is compact.

Note

• Theorem (\(\mathbb{R}^k\)): For any \(E \subset \mathbb{R}^k\), the following are equivalent:

  1. \(E\) is closed and bounded.

  2. \(E\) is compact.

  3. \(E\) is sequentially compact.

Note

• Theorem (\(\mathbb{R}^k\)): Every bounded, infinite subset of \(\mathbb{R}^k\) has a limit point in \(\mathbb{R}^k\) [Bolzano-Weierstrass].

Perfect Sets

Note

• Theorem: If \(P\) is a nonempty, perfect set in \(\mathbb{R}^k\), then \(P\) is uncountable.

Note

• Corollary: Every interval \([a,b]\) is uncountable.

Connected Sets

Note

• Separated Sets: \(A,B \subset X\) are separated if

  \[A \cap B^{\circ} = \varnothing \ \wedge\ A^{\circ} \cap B = \varnothing.\]

Tip

Cannot hop off of one and hop on to another without stepping in something in between.

Note

• Connected Set: \(E \subset X\) is connected if it is not a union of nonempty separated sets.

Note

• Theorem: For a set \(E \subset \mathbb{R}\),

  \[E \text{ is connected} \Longleftrightarrow \bigl(\forall x,y \in E,\ (x<z<y) \Longrightarrow z \in E\bigr).\]