[EE-S] Kyoto Films Developed

 So, I've developed my glorious 5 rolls of films after finishing my trip. The pics were well developed than I thought, but I guess some kind of light leak is happening. A friend of mine told me that he would take care of it, thanks to him. So from the post before, I bought my first film camera at Kyoto and I took 5 rolls of films through my journey. It's about 220 shots, and as I'm a beginner using film cameras, I thought it would be OK for only about a half to be well developed, but it turned out to be that most of them (except the light exposed ones) developed in a good shape. I chose some of the best shots (in my opinion) from Kyoto rolls.

1/20 Kinkakuji

 Right after buying my camera, we headed to Kinkakuji, a marvelous temple wrapped with gold foil, which was my first sight to capture. 

 To ride a bus to Kyoto station, we headed towards the bus station in front of McDonald's and before the sun sets, I took some pictures of Kyoto streets. Kyoto gives an idyllic aesthetic, unlike from cities like Tokyo which rather gives urban aesthetics. As sunshine slowly disappeared from the sky, I stopped taking pics as the camera wasn't suitable for the low light environment. 

1/21 Fushimi-Inari Jinjya ~ Ginkakuji

 One of the sceneries of Kyoto I really wished to see was the aesthetically aligned Toriis from Fushimi-Inari Jinjya, but I was too lazy to wake up early. As tourists rush like a flood around 10am, I had to wake up early and get to there as fast as I could, thanks to Glenfiddich I kind of overslept. Well, that doesn't mean that I didn't enjoy the scenery but there were way too many heads in every shot.

 Even though it was a rush of tourists, I still enjoyed this place, but I still always think about it would be better if I had been there earlier. I ran out of shots so I changed to a new film roll.

 I took some pics of rails and trains, as I like them a lot. These are the pics of the Keihan line while heading to Kiyomizudera. 

 Kyoto has lots of shrines and temples, and this day we've been to like 5 or more of them. The next stop was Kiyomizudera, also a famous tourist attraction in Kyoto. 

 But more than Kiyomizu-dera, I was really looking forward to seeing the streets of Kyoto around Kiyomizu-dera. 

 Kyoto really gives an antique view, as if time has stopped from the Edo period. We walked a lot, about 25 kilometers this day. We headed to Yasaka Jinjya, again another shrine. 

We haven't spent a lot of time here, took some pics and headed right to Heian Jingu. 

 After having lunch ( we really needed some time to rest our legs and feet, they were literally burning ), I bought another film roll from a local film store. After sightseeing Heian Jingu we headed to Ginkakuji, but this time we took a bus. We stopped at the 'Philosopher's Path', a path along a small streamside. Before that, we've been to Ginkakuji.

 Taking some break at a coffee shop around the Philosopher's Path, we took some shots of this rustic path. Personally, I love this shot the most. Gives kind of a Showa vibe, and the winter flowers with the bicycle look really gorgeous. It looks like a postcard from 60's. 

 Walking around for about 25 kilometers, we were knocked out. Before having dinner with some beer, I stopped off at some record stores around the city hall. I'll talk about them in later posts. Well, so these are some of the shots that weren't burnt from light leakage. 

[Top] Different Definitions of Topology

 Many topology texts introduce general topological spaces with the defintion using open sets and its condition for being an open set. But we can define a topology of a certain set in differen Topological space as a mathematical object is merely a 'pair', $(X,\mathcal{T})$ equipped with a set $X$ and a collection  $\mathcal{T}\subseteq \mathcal{P}(X)$ which we call the 'topology', that satisfies some conditions. Now $\mathcal{T}$ defines the topology, and using open sets, closed sets, neighborhoods, closures, and interiors we can define in different 'words' but all equivalent.

 As I mentioned, most of the topology texts define a topology of set by indtroducing open sets, so defining topology by open sets are familiar to us. We define a 'collection of open sets' of a given  set, $\mathcal{T}\subseteq \mathcal{P}(X)$ and we call the elements of this collection open sets. $\mathcal{T}$ should satisfy, 

$$ \begin{split} \emptyset ,X &\in \mathcal{T} \\ \mathcal{S} \subseteq \mathcal{T} &\Rightarrow \cup \mathcal{S} \in \mathcal{T} \\ U,V \in \mathcal{T} &\Rightarrow \text{ } U \cap V \in \mathcal{T} \end{split}$$

 Now using a similar object known as the closed sets, we can define a topology in a slightly different way. Closed sets are compliments of open sets, so we define a 'collection of closed sets' $\mathcal{C} \subseteq \mathcal{P}(X)$ and the elements of this collection are closed sets. $\mathcal{C}$ should satisfy,

$$ \begin{split} \emptyset ,X &\in \mathcal{C} \\ \mathcal{S} \subseteq \mathcal{C} &\Rightarrow \cap \mathcal{S} \in \mathcal{C} \\ U,V \in \mathcal{T} &\Rightarrow \text{ } U \cup V \in \mathcal{C} \end{split}$$

 The equivalence of the first two definitions seem trivial by the nature of open sets and closed sets. As they are defined by the compliment of each other, it seems trivial. And it also makes $\emptyset$ and $X$ a trivial 'clopen' set. We call the nontrivial clopen sets as proper clopen sets, and whether there exists a proper clopen set is related with the connectedness of the given topological space. ( will be continued in later posts )

  We can also define a topology using neighborhoods. A neighborhood is defined by a mapping $\mathcal{N}:X\rightarrow\mathcal{P}(\mathcal{P}(X))$ which maps $x \in X$ as $x \mapsto \mathcal{N}_{x}$. We call $\mathcal{N}_{x}$ as the neighborhood of $x$ when it satisfies the following conditions.

$$\begin{split} \forall x \in X : N\in \mathcal{N}_{x} &\Rightarrow x \in N \\ N\in\mathcal{N}_{x} \land N\subseteq S \subseteq X &\Rightarrow S\in\mathcal{N}_{x} \\ M,N\in\mathcal{N}_{x} &\Rightarrow M\cap N \in \mathcal{N}_{x} \\ N \in \mathcal{N}_{x} &\Rightarrow \exists\mathcal{N}_{x} \text{ } s.t \text{ } \forall y\in M, N\in\mathcal{N}_{y} ,M\in\mathcal{N}_{x}\end{split}$$

 Now considering closures and interiors, someone may ask "How can we think of closures and interiors when we even don't have the notion of open/closed sets?" The answer to this question is to regard closures and interiors as a mapping, $cl$ and $int$ which makes the pair $(X,cl)$ and $(X,int)$ a topological space. The mappping $cl:\mathcal{P}(X)\rightarrow\mathcal{P}(X)$ is defined with below conditions.

$$\begin{split} cl\text{ }\emptyset &= \emptyset \\ \forall A \subseteq X &\Rightarrow A \subseteq cl \text{ } A \\ \forall A,B\subseteq X &\Rightarrow cl(A\cup B)=cl(A)\cup cl(B) \\ \forall A\subseteq X &\Rightarrow cl(cl\text{ }A)=cl\text{ }A \end{split}$$

 The mapping $int:\mathcal{P}(x)\rightarrow\mathcal{P}(x)$ is defined with below conditions.

$$\begin{split} int\text{ }X &= X \\ \forall A \subseteq X &\Rightarrow A \supseteq int \text{ } A \\ \forall A,B\subseteq X &\Rightarrow int(A\cap B)=int(A)\cap int(B) \\ \forall A\subseteq X &\Rightarrow int(int\text{ }A)=int\text{ }A \end{split}$$

 Citing nLab, the definitions mentioned above can be written again as, a set with a frame of open sets, a set with a co-frame of closed sets satisfying the dual axioms, a set with any collection of subsets, a pair $(X,int)$ where $int$ is a left exact comonad on $\mathcal{P}(X)$, and a pair $(X,cl)$ where $cl$ is a right exact Moore closure operator satisfying the dual of $int$.