Let \(X:=\overline{\mathbb{D}} \subseteq \mathbb{R^2}\) (Closed Unit Disc) and let \(Y:=\mathbb{S^2} \subseteq \mathbb{R^3}\) (Unit Sphere). Verify the following:

(1) If \(z,w \in \mathbb{T}\)then set \(z \in w\). If \(z,w \in \mathbb{D}\) then \(z \in w\) provided \(z=w\). Then \(\sim\) defines an equivalence relation on \(X\).

(2) \(f:X \rightarrow Y,f(tx,ty)=(cos\pi(1-t),xsin\pi(1-t),ysin\pi(1-t))\) for \(0 \le t \le 1,(x,y) \in \mathbb{T}\) is a continuous surjection.

Conclude that \(X/ \sim\) and \(Y\) are homeomorphic.

Find one-point compactifications of \(\mathbb{R}\) and \(\mathbb{R^2}\) .

Let \(\mathbb{R^{n+1}} \backslash \{0\}\) be the punctured Euclidean space and let \(\mathbb{S^n}\) denote the unit sphere in \(\mathbb{R^{n+1}}\). Define \(g : \mathbb{R^{n+1}} \backslash \{0\} \rightarrow \mathbb{S^{n}}\) by \(g(x)=x \backslash ||x||_2\). Show that \(g\) is an open map.

Show that \(\mathbb{P^1}(\mathbb{R})\) is homeomorphic to the unit circle \(\mathbb{T}\).

The space \(\mathbb{Q}\) is not connected. In fact, for any irrational \(x \in \mathbb{R}\), the set \([-x,x] \cap \mathbb{Q}= (-x,x) \cap \mathbb{Q}\) is both open and closed in the subspace topology on \(\mathbb{Q}\).

If \(A\) is connected and \(B\) is a set such that \(A \subseteq B \subseteq \overline{A}\) then \(B\) is also connected.

 Show that the complement of any countable subset \(C\) in \(\mathbb{R^2}\) is path-connected.

 Show that the continuous image of a path-connected space is path-connected.

For open subset \(U\) of \(\mathbb{R^n}\) , show that \(U\) is connected if and only if \(U\) is path-connected.

Let \(X\) be a compact space with a nested sequence \(\{C_n\}\) of non-empty closed subsets: \(C_1 \supseteq C_2 \supseteq C_3 \cdots\) . Show that the intersection \(\cap_nC_n\) is non-empty.

A topological space \(X\) is compact if and only if for every collection \(C\) with finite intersection property, \(\cap_{C \in C}C\) is non-empty.