Comb space
Overview
 
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a comb space is a subspace
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...

 of that looks rather like a comb
Comb
A comb is a toothed device used in hair care for straightening and cleaning hair or other fibres. Combs are among the oldest tools found by archaeologists...

. The comb space has some rather interesting properties and provides interesting counterexamples. The topologist's sine curve
Topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example....

 has similar properties to the comb space. The deleted comb space is an important variation on the comb space.
Consider with its standard topology and let K be the set .
Quotations

It may be necessary to use methods other than constitutional ones.

"ZAPU deposes Mr. Nkomo as Leader", The Times, 9 July 1962, p. 9.

Our votes must go together with our guns. After all, any vote we shall have, shall have been the product of the gun. The gun which produces the vote should remain its security officer – its guarantor. The people's votes and the people's guns are always inseparable twins.

Martin Meredith, "Our Votes, Our Guns: Robert Mugabe and the Tragedy of Zimbabwe".

We are still exchanging blows with the British government. They are using gay gangsters. Each time I pass through London, the gangster regime of Blair

expresses its dismay'.

What we hate is not the color of their skins but the evil that emanates from them.

Speech in Harlem, New York (September 2000), quoted in Michael Radu, "State of Disaster", National Review, 27 May 2002

Our party must continue to strike fear in the heart of the white man, our real enemy!

"Whites are real enemy, warns Mugabe", Irish Times, 15 December 2000, p. 11.

The white man is not indigenous to Africa. Africa is for Africans. Zimbabwe is for Zimbabweans.

ibid.

We have fought for our land, we have fought for our sovereignty, small as we are we have won our independence and we are prepared to shed our blood…. So, Blair keep your England, and let me keep my Zimbabwe.

Speech at the Earth Summit in Johannesburg (2 September 2002), quoted in John Battersby and Andrew Grice, "Anti-West anger at summit as Mugabe rounds on Blair", The Independent, 3 September 2002, p. 1.

Let Blair and the British government take note and listen. Zimbabwe is for Zimbabweans. Our people are overjoyed, the land is ours. We are now the rulers and owners of Zimbabwe.

Michael White, Andrew Meldrum, "Commonwealth leaders delay decision on defiant Mugabe", The Guardian, 6 December 2003, p. 2.

 
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