Understanding Topology of simplical complexes

Question

Context

I'm trying to understand the Topology of the simplical complexes as explained in this wiki,

First, define $|K|$ as a subset of $[0,1]^{S}$ consisting of functions $t: S \rightarrow[0,1]$ satisfying the two conditions: $$ \begin{aligned} &\left\{s \in S: t_{s}>0\right\} \in K \\ &\sum_{s \in S} t_{s}=1 \end{aligned} $$ Now think of the set of elements of $[0,1]^{S}$ with finite support as the direct limit of $[0,1]^{A}$ where $A$ ranges over finite subsets of $S$, and give that direct limit the induced topology. Now give $|K|$ the subspace topology.

I'm having trouble understanding the words 'finite support' and 'direct limit'.

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Research

Finite support

I found this SE answer more or less easy to comprehend,

"It is convenient to say that a function that vanishes outside a set of finite measure has finite support and define its support to be $\{x: \ f(x) \neq 0\}$".

My issues with this are:

1. In the text from wiki , they say a set of elements with finite support.. but in the above it's about functions?
2. Secondly, how can I think of finite measure? I haven't learned of measure theory yet. From reading this wiki, I am guessing the idea is a set of finite n-volume?

Direct limit

I more or less understood the idea given in "Direct limits of algebraic object" in the wiki here but I am having difficulties making it useful here.

1. What does the unit interval raised to a set mean? (see first quoted text, $\left[ 0 ,1 \right]^A$ is mentioned)
2. I don't see any $f_{ij}$ or the Indexing set as shown in wiki , so it is not clear to me how if the mentioned section actually applies. How do I make it apply?
3. If it is the wrong version of the definition that I am using, how can I understand what the direct limit means here?

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Closing remarks and background

Could someone helped me with these questions , and if possible, give an intuitive explanation how the topology of simplicial complexes actually works in simple words?

I'm reading a Physics book and Simplicial Complexes was one of the mentioned ideas. I thought it was interesting and am trying to get more intuition for it.

Answer 1

$[0,1]^S$ is the set of functions with domain $S$ with values in the set $[0,1]$. Saying that an element of that set has finite support is thus the same as saying it is a function of finite support, i.e. that it vanishes outside a set of finite measure.

A set having finite measure in this context means that it is finite. So a function having finite support, i.e. vanishing outside a set of finite measure, means that that the subset of the domain on which the function has a non-zero value is finite.

The set of fininte subsets of a set $A$, ordered under inclusion, is a directed partially ordered set: any finite subset is contained in itself and for any two finite subsets there is a finite subset containing both of them. Given finite subsets $I\subseteq J\subseteq A$, define $f_{IJ}\colon[0,1]^I\to[0,1]^J$ by $f_{IJ}(g)(x)=\begin{cases}g(x)&x\in I\newline 0&\text{otherwise}\end{cases}$ for $g\colon I\to[0,1]$. Evidently, $f_{JK}\circ f_{IJ}=f_{IK}$ and $f_{II}=\mathrm{id}_{[0,1]^I}$. Thus $f_{IJ}$ define a directed system.

The directed limit of the above directed system can be identified with the set of functions $A\to[0,1]$ that have finite support.

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An abstract simplicial complex is a collection of finite subsets of a set $S$, called abstract simplices, with the property that any subset of a finite subset in the collection is also in the collection.

Every point of the geometric realization $K$ of a simplicial complex can be written as a convex linear combination of elements of an abstract simplex, and uniquely so as a combination with non-zero coefficients.

In other words, $K$ may be identified with $[0,1]$-valued functions on the set $S$ of vertices, whose support (elements of the domain with non-zero values) is an abstract simplex, and whose total sum is $1$.

An open set in the topology is explicitly given by collections of open subsets $U_I\subseteq[0,1]^I$ for each abstract simplex $I$ satisfying $U_I=f_{IJ}^{-1}(U_J)$. In the case where each abstract simplex is contained in a maximal simplex, this is the same as giving a family of open subsets, one for each maximal simplex, that agree on their intersections.