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strongly connected spaces - National University of Singapore
strongly connected spaces - National University of Singapore

... between f(a) and f(b), then there exists an element c ∈ [a, b] such that f(c) = r. This theorem is used in a number of places, for instance when constructing inverse functions such as sin-1(x). The property of the space [a, b] on which the Intermediate Value Theorem depends is the connectedness, and ...
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a note on weakly separable spaces

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... 1. A is an sCI -set and a semi∗ -I-open set in X. 2. A = L ∩ cl(int∗ (A)) for a semi-I-open set L. Proof. (1) ⇒ (2): Suppose that A is an sCI -set and a semi∗ -I-open set in X. Since A is sCI -set, then we have A = L ∩ M, where L ∈ SIO(X) and M is a pre-I-closed set in X. We have A ⊆ M, so cl(int∗ ( ...
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... PROOF. Let x E X* and h.t .s lc ;t net in X* such that every universal subnet of s converges tox inX* Supposex isinX. Let u bcauniversalsubnet ofs. Sinceu-x, uis frequently in X and ulX x. By Lemma 1.3, u is eventually in X. According to Theorem 1.4, s is frequently in X. Let v bc a univcrsd subnct ...
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Continuous function

In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.
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