By K.P. Hart, Jun-iti Nagata, J.E. Vaughan

This e-book is designed for the reader who desires to get a basic view of the terminology of normal Topology with minimum effort and time. The reader, whom we think to have just a rudimentary wisdom of set thought, algebra and research, could be capable of finding what they wish in the event that they will competently use the index. although, this publication comprises only a few proofs and the reader who desires to research extra systematically will locate sufficiently many references within the book.

Key features:

• extra phrases from basic Topology than the other publication ever released • brief and informative articles • Authors contain nearly all of most sensible researchers within the box • broad indexing of terms

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**Extra resources for Encyclopedia of General Topology**

**Sample text**

A space containing it as a subspace). For example, a Tychonoff space X is a subspace ˇ of its Stone–Cech compactiﬁcation. Every separable metric space is a subspace of the Hilbert cube. Throughout this article, X will denote a topological space and Y a subspace of X. 1. Elementary relative separation properties In order to understand the concepts of relativity, we start with the notion of relatively Hausdorff. A subspace Y is Hausdorff in (respectively strongly Hausdorff in) X, if for each distinct two points y1 , y2 ∈ Y (respectively y1 ∈ Y and y2 ∈ X), there are disjoint open sets U1 , U2 of X with yi ∈ Ui (i = 1, 2).

There exist a normal space X and a subspace Y of X such that dim(Y, X) = dim(Y ) and dim(Y, X) = dim(X) (see [11]). Since the inequality dim(Y1 , X1 ) dim(Y2 , X2 ) holds for Y1 ⊂ Y2 ⊂ X1 ⊂ X2 , we have dim(Y, X) dim(Y ) and dim(Y, X) dim(X). Although equality does not hold in general, the following are sufﬁcient conditions for dim(Y, X) = dim(Y ): (a) Y is closed, or (b) Y is a normal subspace in itself and an Fσ -set in X, or (c) Y is C ∗ -embedded in X [6]. It is also known that if Y is a normal subspace in itself and z-embedded in X [MN, p.

Every L-space determines a ﬁlter convergence as follows: x ∈ lim F if and only if there exists a sequence (xn ) that converges to x such that F contains the ﬁlter generated by (xn ). Filter convergences determined by L-spaces are Hausdorff (due to the condition (S0)). Let X be a topological space. A ﬁlter F on X converges to a point x ∈ X if every neighbourhood of x belongs to F . This special ﬁlter convergence satisﬁes (F1), (F2) and also {lim F : F ∈ F} ⊂ lim {F : F ∈ F} for every F ⊂ F(X), and (F4) lim F ⊂ lim Φ(F ) for every F ∈ F(X) and each map Φ : X → F(X) such that x ∈ lim Φ(x) for every x ∈ X.