The sequence of sets: $Z_n=Y_n \cap(X \backslash W), n=1,2,3, \ldots$ are closed. If $Y_n⊈W$ for all $n$, then $Z_n≠∅$ for all $n$. By (a) $Z_∞=Y_∞∖W≠∅$
Assume $Y_∞$ is disconnected by $U,V$. $Y_∞∩U∩V=∅$ $Y_∞∩U≠∅$ $Y_∞∩V≠∅$ By (b) $Y_n⊂U∪V$ for some $n$. wlog $Y_n⊂U$ $Y_∞⊂Y_n⊂U$
⇒ is obvious (composition of continuous functions) ⇐ It suffices to check that $f^{-1}(U×V)$ is open for $U$ open in $X$, $V$ open in $Y$ as $U×V$ is a basis for topology of $X×Y$. Note $U×V=(U×Y)∩(X×V)=p_X^{-1}(U)∩p_Y^{-1}(V)$ $f^{-1}(U×V)=(p_X∘f)^{-1}(U)∩(p_Y∘f)^{-1}(V)$ is open
Take $[0,1]→X×Y,t↦(γ^x,γ^y)$
Pick $y∈L$, $(x,y)∈K×L$, so ∃ neighborhood $U_x^y×V_x^y$ of $(x,y)⊂W$ [as $W$ open & products of open sets form a basis] $\{V_x^y\}_{y∈L}$ covers $L$. ∃ finite $I$ $\{V_x^{y_i}\}_{i∈I}$ covers $L$. Set $U_x=\bigcap_{i∈I}U_x^{y_i},V_x=\bigcup_{i∈I}V_x^{y_i}$
A triangulation of a space $X$ is a simplicial complex $K$ together with a choice of homeomorphism $|K| → X$.
Triangulation: Main pitfall: Need to make sure that each simplex is uniquely determined by its vertices.
5.2. in Lecture notes
Let $N_3$ be the surface obtained from the word $x x y y z z$ (that is we label the boundary of a hexagon by this word and we identify the sides with the same label). Show that we may obtain $N_3$ by identifying the sides of two squares $P_1, P_2$ as described above.Corollary 5.14. The surface $N_h$ is obtained from a 2-sphere by adding $h$ crosscaps. $xxyy$ is the Klein bottle.