Topology.sage Use Sage Math and Set function/module to check if a set is a topology and other implementations for Topology

topology.sage

## topology.sage
## This is my implementation of Topology
##   utilizing
## Sage Math
##  from 
## 
## The main reference that I'll liberally copy from is from is 
##   Lecture 1: Topology (International Winter School on Gravity and Light 2015)
##   Frederic P Schuller
##   The WE-Heraeus International Winter School on Gravity and Light 2015
##
## and http://people.math.sfu.ca/~jtmulhol/math302/notes/2-SetTheory.pdf
## Lecture 2: A Bit of Set Theory
## Jamie Mulholland, Spring 2010 Math 302
## 
################################################################################
## Copyleft 2015, Ernest Yeung <ernestyalumni@gmail.com>
## 
## 20150524
## 
## This program, along with all its code, is free software; you can redistribute 
## it and/or modify it under the terms of the GNU General Public License as 
## published by the Free Software Foundation; either version 2 of the License, 
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## 
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emptyset = Set([])  # cf. http://doc.sagemath.org/html/en/reference/structure/sage/sets/set.html an_element()

# cf. Exercise 2: Topologies on a simple set, Topology tutorial sheet, The WE-Heraeus International Winter School on Gravity and Light 2015

M = Set(x for x in range(1,5))

O_1 = Set( [ emptyset, Set([1]), M ] )
O_2 = Set( [ emptyset, Set([1]), Set([2]), M ])

# this is an example of checking Axiom 1 of the definition of a topology, that the empty set is in the topology
emptyset in O_1 # True


def chaotictopology(set):
    """
    chaotictopology = chaotictopology(set)
    Returns a Set
    chaotic topology is also known as trivial topology or indiscrete topology (all the same thing)
    """
    return Set([emptyset, set])

def Axiom2check(topology):
    """
    Axiom2check = Axiom2check(topology)
    checks Axiom 2, the finite intersection axiom for topology

    e.g. Example of Usage
    Axiom2check(O_1) # True
    """
    
    for U in topology:
    	for V in topology:
	    if U.intersection(V) not in topology:
	       print "This did not satisfy Axiom 2: ", U.intersection(V)
	       return False
    return True


def Axiom3check(topology):
    """
    Axiom3check = Axiom3check(topology)
    checks Axiom 3, that the union of any subset of a topology is in the topology    

    e.g. Example of Usage
    Axiom3check(O_1)
    """
    P = topology.subsets() # power set
    for subset in P.list():
    	if subset != emptyset:
	   if subset.cardinality() > 1:
	      union_alpha = subset[0]
	      for U in subset:
	      	  union_alpha = union_alpha.union(U)
	      if union_alpha not in topology:
	      	 print "This did not satisfy Axiom 3: ", union_alpha
	      	 return False
    return True


def checkcont(targettopology,domaintopology,invfunc):
    for V in targettopology:
    	preimg = []
	for element in V:
	    preimg.append(invfunc(element))
	preimg = Set(preimg)
	if preimg not in domaintopology:
	   print "This V in the target topology makes the function not continuous: ", V
	   return False
    return True

####################################################################################################
# So going back to Exercise 2: Topologies on a simple set, Topology tutorial sheet, The WE-Heraeus International Winter School on Gravity and Light 2015
####################################################################################################

# Question: Does O1 := {∅, {1}, {1, 2, 3, 4}} constitute a topology on M ? 
# Answer: Yes since
emptyset in O_1
Axiom2check(O_1) # True
Axiom3check(O_1) # True
    
# Question: What about O2 := {∅, {1}, {2}, {1, 2, 3, 4}}?
# Answer: No since
emptyset in O_2
Axiom2check(O_2) # True
Axiom3check(O_2) # False

####################################################################################################
# Topology tutorial sheet, The WE-Heraeus International Winter School on Gravity and Light 2015
# cf. Exercise 3: Continuous functions
####################################################################################################

# Question: Let M = {1, 2, 3, 4} and consider the identity map idM : M → M defined by idM(1) = 1, idM(2) = 2, idM(3) = 3, idM(4) = 4.
# Is the map idM continuous if the domain is equipped with the chaotic topology and the target with the topology Otarget := {∅, {1}, {1, 2, 3, 4}}?
# Answer: No

domain31 = chaotictopology(M)
target31 = Set( [ emptyset, Set([1]), M ])

f = lambda(x) : x

checkcont(target31,domain31,f) # False

# Question: Consider the inverse id−1 : M → M of the identity map idM , such that now the target is M
# equipped with the chaotic topology and the domain with the topology {∅, {1}, {1, 2, 3, 4}}. Provide the values of the map id−1 and decide whether id−1 is continuous!
# Answer: Yes

checkcont(domain31, target31, f) # True