this is a topology question:
For these topologies $T_{disc}, T_{codisc}, T_{fin}, T_{count}, T_{st}, T_K, T_{up}, T_{uplim}$ on [0, 1]:
i) find the connected component of $\frac{1}{2}$ in [0, 1], except for $T_K$.
ii) say if [0, 1] is compact. You can assume ([0, 1],Tst) is compact.
iii) say if K := {$\frac{1}{n+1}$| n ∈ N} is compact.
Definitions:
Discrete topology on X: is defined by letting every subset of X be open (and hence also closed), and X is a discrete topological space if it is equipped with its discrete topology;
Co-discrete topology:= {∅, X} defines a topology on X,
Finite topology n := {∅} ∪ {A ⊆ X | X \ A is finite} defines a topology on X,
Countable topology:= {∅} ∪ {A ⊆ X | X \ A is countable} defines a topology on X,
Standard topology:=The collection $B_{st}$ := {(a, b)} defines a basis for a topology on R, called the standard topology,
K topology := The collection $B_{K} := B_{st}$ ∪ {(a, b) \ K}, where K := {$\frac{1}{n}∈ N_{>0}$} defines a basis for a topology on R,
Upper topology:= {∅, R}∪{(a, ∞) | a ∈ R} defines a topology on R,
Upper limit topology:= {(a, b]} defines a basis for a topology on R.
Here is what I have:
i) I'm not sure how to do this. If I have to give an answer, then for $T_st$, I think we can only have singletons {$\frac{1}{2}$}, as its own connected components? or {$\frac{1}{2}$} $\cup$ {some singleton}. Because $T_st$ defines the interval (a,b), and it's impossible to find an closed interval that fulfills the requirement of connected space. For instance, take the interval (0,$\frac{1}{2}$) and assume it's connected. We can rewrite it as (([0,$\frac{1}{4}$))∪([$\frac{1}{4}$,$\frac{1}{2}$]), since $\frac{1}{4}$∈([$\frac{1}{4}$,$\frac{1}{2}$])≠ϕ , it follows that [0,$\frac{1}{4}$)=ϕ. Therefore, if c∈(0,$\frac{1}{2}$), then c∉[0,$\frac{1}{4}$), and c is greater than or equal to $\frac{1}{4}$. And we can make c≤a because it's arbitrary, thus a=c, and the only connected part is singletons.
Please correct me if I'm wrong, and I don't know how to work this for $T_{disc}, T_{codisc}, T_{fin}, T_{count}, T_K, T_{up}$ (If the above is correct, I think $ T_{uplim}$ will follow something similar.)
ii) Given: Let (X, T ) be a space, and Y ⊆ X a subset. • The space (Y, $T_Y$ ) is compact if and only if any covering of Y by open subsets in X contains a finite subcollection covering Y . • If Y ⊆ Z ⊆ X, (Y, $T_Y$ ) is compact if and only if (Y,($T_Z)_Y$ ) is compact.
For $T_K, T_{up}, T_{uplim}$, I not so sure how to find the finite covering, the idea is too abstract for me. I guess for $T_K$, I think K := {$\frac{1}{n}$} is finite and intuitively, the incountable $\frac{1}{n}$ will cover the entire interval, except at point 0 because it is a limit point.
iii) Similarly, here it's more direct that at 0, we encounter a special point, but I'm still not sure about how the argument should go.
Thank you in advance, any help is appreciated.