What is topology?

Jon Woolf

February, 2010 A H R Acton Town Hainault Ravenscourt Park Aldgate Hammersmith Rayners Lane Aldgate East Hampstead Redbridge Alperton Hanger Lane Regents Park Amersham Harlesden Richmond Angel Harrow-on-the-Hill Rickmansworth Archway Hatton Cross Roding Valley Arnos Grove Heathrow Rotherhithe Arsenal Hendon Central Royal Albert B High Barnet Royal Oak Baker Street High Street Kensington Royal Victoria Balham Highbury & Islington Ruislip Bank Highgate Ruislip Gardens Barbican Hillingdon Ruislip Manor Barking Holborn Russell Square Barkingside Holland Park S Barons Court Holloway Road Seven Sisters Bayswater Hornchurch Shadwell Becontree Hounslow Central Shepherds Bush Belsize Park Hounslow East Shoreditch Bermondsey Hounslow West Snaresbrook Bethnal Green Hyde Park Corner South Ealing Blackfriars I South Harrow Blackhorse Road Ickenham South Kensington Bond Street K South Kenton Borough Kennington South Quay Boston Manor Kensal Green South Ruislip Bounds Green Kensington (Olympia) South Wimbledon Bow Church Kentish Town South Woodford Bow Road Kenton Southfields Brent Cross Kew Gardens Southgate Brixton Kilburn Southwark Bromley-by-Bow Kilburn Park St. James’s Park Buckhurst Hill Kings Cross St. Pancras St. Johns Wood Burnt Oak Kingsbury St. Pauls C Knightsbridge Stamford Brook Caledonian Road L Stanmore Camden Town Ladbroke Grove Stepney Green Canada Water Lambeth North Stockwell Canary Wharf Lancaster Gate Stonebridge Park Canning Town Latimer Road Stratford Cannon Street Leicester Square Sudbury Hill Canons Park Leyton Sudbury Town A H R Chalfont & Latimer Leytonstone Surrey Quays Acton Town Hainault Ravenscourt Park Chalk Farm Limehouse Swiss Cottage Aldgate Hammersmith Rayners Lane Chancery Lane Liverpool Street T Aldgate East Hampstead Redbridge Charing Cross London Bridge Temple Alperton Hanger Lane Regents Park Chesham Loughton Theydon Bois Amersham Harlesden Richmond Chigwell M Tooting Bec Angel Harrow-on-the-Hill Rickmansworth Chiswick Park Maida Vale Tooting Broadway Archway Hatton Cross Roding Valley Chorleywood Manor House Tottenham Court Rd Arnos Grove Heathrow Rotherhithe Clapham Common Mansion House Tottenham Hale Arsenal Hendon Central Royal Albert Clapham North Marble Arch Totteridge & Whetstone B High Barnet Royal Oak Clapham South Marylebone Tower Gateway Baker Street High Street Kensington Royal Victoria Cockfosters Mile End Tower Hill Balham Highbury & Islington Ruislip Colindale Mill Hill East Tufnell Park Bank Highgate Ruislip Gardens Colliers Wood Monument Turnham Green Barbican Hillingdon Ruislip Manor Covent Garden Moor Park Turnpike Lane Barking Holborn Russell Square Croxley Moorgate U Barkingside Holland Park S D Morden Upminster Bridge Barons Court Holloway Road Seven Sisters Dagenham East Mornington Crescent Upney Bayswater Hornchurch Shadwell Dagenham Hthway Mudchute Upton Park Becontree Hounslow Central Shepherds Bush Debden N Uxbridge Belsize Park Hounslow East Shoreditch Devons Row Neasden V Bermondsey Hounslow West Snaresbrook Dollis Hill New Cross Vauxhall Bethnal Green Hyde Park Corner South Ealing E New Cross Gate Victoria Set of tube stations Blackfriars I South Harrow Ealing Broadway Newbury Park W Blackhorse Road Ickenham South Kensington Ealing Common North Acton Walthamstow Central Bond Street K South Kenton Earls Court North Ealing Wanstead Borough Kennington South Quay East Acton North Greenwich Wapping Boston Manor Kensal Green South Ruislip East Finchley North Harrow Warren Street Bounds Green Kensington (Olympia) South Wimbledon East India North Wembley Warwick Avenue Bow Church Kentish Town South Woodford East Putney Northfields Waterloo Bow Road Kenton Southfields Eastcote Northolt Watford Brent Cross Kew Gardens Southgate Edgware Northwick Park Wembley Central Brixton Kilburn Southwark Edgware Road Northwood Wembley Park Bromley-by-Bow Kilburn Park St. James’s Park Elephant & Castle Northwood Hills West Acton Buckhurst Hill Kings Cross St. Pancras St. Johns Wood Elm Park Notting Hill Gate West Brompton Burnt Oak Kingsbury St. Pauls Embankment O West Finchley C Knightsbridge Stamford Brook Epping Oakwood West Ham Caledonian Road L Stanmore Euston Old Street West Hampstead Camden Town Ladbroke Grove Stepney Green Euston Square Osterley West Harrow Canada Water Lambeth North Stockwell F Oval West India Quay Canary Wharf Lancaster Gate Stonebridge Park Fairlop Oxford Circus West Kensington Canning Town Latimer Road Stratford Farringdon P West Ruislip Cannon Street Leicester Square Sudbury Hill Finchley Central Paddington Westbourne Park Canons Park Leyton Sudbury Town Finchley Road Park Royal Westferry A H R Chalfont & Latimer Leytonstone Surrey Quays Finsbury Park Parsons Green Westminster Acton Town Hainault Ravenscourt Park Chalk Farm Limehouse Swiss Cottage Fulham Broadway Perivale White City Aldgate Hammersmith Rayners Lane Chancery Lane Liverpool Street T G Piccadilly Circus Whitechapel Aldgate East Hampstead Redbridge Charing Cross London Bridge Temple Gants Hill Pimlico Willesden Green Alperton Hanger Lane Regents Park Chesham Loughton Theydon Bois Gloucester Road Pinner Willesden Junction Amersham Harlesden Richmond Chigwell M Tooting Bec Golders Green Plaistow Wimbledon Angel Harrow-on-the-Hill Rickmansworth Chiswick Park Maida Vale Tooting Broadway Goldhawk Road Poplar Wimbledon Park Archway Hatton Cross Roding Valley Chorleywood Manor House Tottenham Court Rd Goodge Street Preston Road Wood Green Arnos Grove Heathrow Rotherhithe Clapham Common Mansion House Tottenham Hale Grange Hill Prince Regent Woodford Arsenal Hendon Central Royal Albert Clapham North Marble Arch Totteridge & Whetstone Great Portland St Pudding Mill Lane Woodside Park B High Barnet Royal Oak Clapham South Marylebone Tower Gateway Green Park Putney Bridge Baker Street High Street Kensington Royal Victoria Cockfosters Mile End Tower Hill Greenford Q Balham Highbury & Islington Ruislip Colindale Mill Hill East Tufnell Park Gunnersbury Queens Park Bank Highgate Ruislip Gardens Colliers Wood Monument Turnham Green Queensbury Barbican Hillingdon Ruislip Manor Covent Garden Moor Park Turnpike Lane Queensway Barking Holborn Russell Square Croxley Moorgate U Barkingside Holland Park S D Morden Upminster Bridge Barons Court Holloway Road Seven Sisters Dagenham East Mornington Crescent Upney Bayswater Hornchurch Shadwell Dagenham Hthway Mudchute Upton Park Becontree Hounslow Central Shepherds Bush Debden N Uxbridge Belsize Park Hounslow East Shoreditch Devons Row Neasden V Bermondsey Hounslow West Snaresbrook Dollis Hill New Cross Vauxhall Bethnal Green Hyde Park Corner South Ealing E New Cross Gate Victoria Blackfriars I South Harrow Ealing Broadway Newbury Park W Blackhorse Road Ickenham South Kensington Ealing Common North Acton Walthamstow Central Bond Street K South Kenton Earls Court North Ealing Wanstead Borough Kennington South Quay East Acton North Greenwich Wapping Boston Manor Kensal Green South Ruislip East Finchley North Harrow Warren Street Bounds Green Kensington (Olympia) South Wimbledon East India North Wembley Warwick Avenue Bow Church Kentish Town South Woodford East Putney Northfields Waterloo Bow Road Kenton Southfields Eastcote Northolt Watford Brent Cross Kew Gardens Southgate Edgware Northwick Park Wembley Central Brixton Kilburn Southwark Edgware Road Northwood Wembley Park Bromley-by-Bow Kilburn Park St. James’s Park Elephant & Castle Northwood Hills West Acton Buckhurst Hill Kings Cross St. Pancras St. Johns Wood Elm Park Notting Hill Gate West Brompton Burnt Oak Kingsbury St. Pauls Embankment O West Finchley C Knightsbridge Stamford Brook Epping Oakwood West Ham Caledonian Road L Stanmore Euston Old Street West Hampstead Camden Town Ladbroke Grove Stepney Green Euston Square Osterley West Harrow Canada Water Lambeth North Stockwell F Oval West India Quay Canary Wharf Lancaster Gate Stonebridge Park Fairlop Oxford Circus West Kensington Canning Town Latimer Road Stratford Farringdon P West Ruislip Cannon Street Leicester Square Sudbury Hill Finchley Central Paddington Westbourne Park Canons Park Leyton Sudbury Town Finchley Road Park Royal Westferry Chalfont & Latimer Leytonstone Surrey Quays Finsbury Park Parsons Green Westminster Chalk Farm Limehouse Swiss Cottage Fulham Broadway Perivale White City Chancery Lane Liverpool Street T G Piccadilly Circus Whitechapel Charing Cross London Bridge Temple Gants Hill Pimlico Willesden Green Chesham Loughton Theydon Bois Gloucester Road Pinner Willesden Junction Chigwell M Tooting Bec Golders Green Plaistow Wimbledon Chiswick Park Maida Vale Tooting Broadway Goldhawk Road Poplar Wimbledon Park Chorleywood Manor House Tottenham Court Rd Goodge Street Preston Road Wood Green Clapham Common Mansion House Tottenham Hale Grange Hill Prince Regent Woodford Clapham North Marble Arch Totteridge & Whetstone Great Portland St Pudding Mill Lane Woodside Park Clapham South Marylebone Tower Gateway Green Park Putney Bridge Cockfosters Mile End Tower Hill Greenford Q Colindale Mill Hill East Tufnell Park Gunnersbury Queens Park Colliers Wood Monument Turnham Green Queensbury Covent Garden Moor Park Turnpike Lane Queensway Croxley Moorgate U D Morden Upminster Bridge Dagenham East Mornington Crescent Upney Dagenham Hthway Mudchute Upton Park Debden N Uxbridge Devons Row Neasden V Dollis Hill New Cross Vauxhall E New Cross Gate Victoria Ealing Broadway Newbury Park W Ealing Common North Acton Walthamstow Central Earls Court North Ealing Wanstead East Acton North Greenwich Wapping East Finchley North Harrow Warren Street East India North Wembley Warwick Avenue East Putney Northfields Waterloo Eastcote Northolt Watford Edgware Northwick Park Wembley Central Edgware Road Northwood Wembley Park Elephant & Castle Northwood Hills West Acton Elm Park Notting Hill Gate West Brompton Embankment O West Finchley Epping Oakwood West Ham Euston Old Street West Hampstead Euston Square Osterley West Harrow F Oval West India Quay Fairlop Oxford Circus West Kensington Farringdon P West Ruislip Finchley Central Paddington Westbourne Park Finchley Road Park Royal Westferry Finsbury Park Parsons Green Westminster Fulham Broadway Perivale White City G Piccadilly Circus Whitechapel Gants Hill Pimlico Willesden Green Gloucester Road Pinner Willesden Junction Golders Green Plaistow Wimbledon Goldhawk Road Poplar Wimbledon Park Goodge Street Preston Road Wood Green Grange Hill Prince Regent Woodford Great Portland St Pudding Mill Lane Woodside Park Green Park Putney Bridge Greenford Q Gunnersbury Queens Park Queensbury Queensway Geometric map of the tube Topological map of the tube Spaces from sets with relations a

b

c Spaces from sets with relations a

b

c Spaces from sets with relations a

b

c Spaces from sets with relations a

b

c Spaces from sets with relations a

b

c Spaces from sets with relations a

b

c Spaces from sets with relations

a b != c In a sense, all spaces arise in this way.

The points of a space form a set, paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. The points of a space form a set, paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way. paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way.

The points of a space form a set, A deformation of one path into another is a relation between relations. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way.

The points of a space form a set, paths in the space are relations between the start and end points. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way.

The points of a space form a set, paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way.

The points of a space form a set, paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. And so on through higher and higher relations. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way.

The points of a space form a set, paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. The resulting space is ‘equivalent’ to the original one.

Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way.

The points of a space form a set, paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. Spaces as sets with relations, with relations,...

This construction of spaces from sets with relations is very general. In a sense, all spaces arise in this way.

The points of a space form a set, paths in the space are relations between the start and end points. A deformation of one path into another is a relation between relations. And so on through higher and higher relations.

In this way we build a set with relations, and relations between relations and so on ad infinitum out of a space. We can turn this set with relations etc back into a space using the previous construction. The resulting space is ‘equivalent’ to the original one. Homotopy theory Homotopy theory Homotopy theory Homotopy theory Homotopy theory Homotopy theory Homotopy theory Homotopy theory