Topos= place
Logos= study 

• Calculus of spatially-propertied invariants under consistency of 'stretching without tearing or gluing'

• A particular mathematical object studied in this area - a family of open sets which contains the empty set and the entire space, closed under the operations union and finite intersection.

Topology is one of the great unifying ideas of mathematics

1. Shows up naturally in spaces in mathematics, abstract algebra, and geometry.
    
2. Helps to define and elaborate some useful properties of spaces and maps.

• connectedness
• compactness
• continuity

Algebraic topology - the study of 'topological spaces' and the maps between them.

Geometry

• Connectivity properties - what is connected to what - 'The Seven Bridges'
   
• No non-vanishing continuous tangent vector on a sphere - the hairy ball theorum, 'as long as it has no holes'.

So just what properties do these problems rely on

• homeomorphic - 'sameness' for the purposes of topology
   
• visually, two spaces are topologically equivalent if one can be deformed into the other without cutting it apart or gluing pieces of it together

One particular problem- Any definitive'set of points' is not available in toplogical space. Consider instead the lattice of open sets as the basic notion of the theory.

• certain structures are defined on arbitrary categories which allow the definition of sheaves on those categories
   
• leads to the definition of quite general cohomology theories